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Smooth Proper Curves Divisors Genus and Ramification — Examples

1 · Prerequisites

2 · Summary

These computations and counterexamples exercise the curve, divisor, genus and ramification material on explicit examples.

On the projective line the divisors of homogeneous polynomials are computed, the Riemann-Roch spaces $L (d [\infty])$ are identified with polynomials of degree at most $d$ , the complete linear systems $∣ d [\infty] ∣$ with the effective divisors of degree $d$ , and the associated morphisms are recognised as Veronese maps. Two two-dimensional subsystems of $∣ 2 [\infty] ∣$ compare a base-point-free system defining a degree-two morphism with one having a single base point, and a smooth conic with a rational point is parametrised by lines through that point and shown to be isomorphic to the projective line.

The genus computations read the arithmetic and geometric genera off defining equations. A nodal cubic and a cuspidal cubic have arithmetic genus one, and their normalizations, computed explicitly, have genus zero; a smooth plane quartic has genus three. Delta invariants are calculated from the local rings at the singularities, and the divisor degree over a field that is not algebraically closed exhibits the residue-degree weighting. A hyperelliptic curve is presented as the double cover $y^{2} = f (x)$ of the projective line, with its branch points, ramification and genus computed directly.

Ramification is computed for the power maps $[s : t] \mapsto [s^{n} : t^{n}]$ of the projective line, where the indices at the two fixed points and the different are read off from the differential, and the $p$ -th power map in characteristic $p$ shows that the canonical bundle ramification formula fails without a separability hypothesis. Two further boundaries are recorded: a rational map from a singular curve that cannot be extended to a morphism, and a nontrivial degree-zero line bundle with no nonzero section.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-10-02Open item page →

Divisors and complete linear systems on the projective line

Example

Assume the Axiom of Choice for the current divisor, cohomology, and projective-space supplier routes (The Axiom of Choice). On $P_{k}^{1}$ with affine coordinate $t$ and point at infinity $\infty$ (Two-affine projective line and its twists, Relative projective space from standard charts), a nonzero polynomial $p (t)$ of degree $m$ , viewed as a rational function, has divisor $div (p) = Z (p) - m [\infty],$ where $Z (p)$ is the effective divisor of its affine zeros, of degree $m$ ; the Riemann-Roch space $L (d [\infty])$ (The space L(D)) consists exactly of the polynomials of degree at most $d$ together with $0$ . Consequently $\deg_{k} (d [\infty]) = d$ , the space $L (d [\infty])$ has dimension $d + 1$ for $d \geq 0$ and is zero for $d < 0$ , the complete linear system $∣ d [\infty] ∣$ (Complete linear system) is the set of effective divisors of degree $d$ for $d \geq 0$ , and is empty for $d < 0$ . For $d \geq 0$ this set is identified with the set of $k$ -lines in $L (d [\infty])$ , equivalently the $k$ -rational points of the projective scheme of lines $P (L (d [\infty]))$ ; the linear system is a set, while $P_{k}^{d}$ below is a scheme. For $d \geq 1$ , the morphism associated with the base-point-free system $L (d [\infty])$ (A base-point-free linear system defines a morphism to projective space) is the degree- $d$ Veronese closed immersion of schemes $P_{k}^{1} \to P_{k}^{d}$ (The degree-d Veronese map). For $d = 0$ , the single generator $1$ of $L (0)$ gives the constant structure morphism $P_{k}^{1} \to P_{k}^{0}$ , which is not an embedding; for $d < 0$ , $L (d [\infty]) = 0$ and there is no associated projective morphism.

Current supplier interfaces. The current Projective-line curve and divisor basics body gives the closed- point degree and divisor classification used in steps 1.1 and 3.1. The current Invertible sheaf of cartier divisor and the projective-map suppliers in [F4]–[F5] give the sheaf and section construction; the current Cartier and Weil divisors agree on a smooth curve body identifies Cartier and Weil divisors, with its Dependent Choice premise supplied from AC through AC implies DC implies countable choice. These supplier files are present. Their current decisions remain separate from the computations recorded here.

Facts & Assumptions

Given: A field $k$ , the projective line $P_{k}^{1}$ with standard charts $U_{0} = Spec k [t]$ and $U_{1} = Spec k [u]$ , $t u = 1$ , point at infinity $\infty = [0 : 1]$ the pole of $t$ , and an integer $d \in Z$ .

[F1]

The current in-run item Projective-line curve and divisor basics states that $P_{k}^{1}$ is a smooth proper geometrically integral curve of genus $0$ , that a closed point $V (g)$ attached to a monic irreducible $g \in k [t]$ of degree $e$ has residue degree $e$ and $div (g) = [V (g)] - e [\infty]$ , and that every divisor $D$ on $P_{k}^{1}$ is linearly equivalent to $\deg_{k} (D) [\infty]$ ; it also identifies $O (1) ≅ O (\infty)$ .

[F2]

On a smooth curve divisors are finite $Z$ -combinations of closed points, $\deg_{k} (\sum_{x} n_{x} [x]) = \sum_{x} n_{x} [κ (x) : k]$ , effectivity is nonnegativity of all coefficients, the order function ${ord}_{x}$ is additive with ${ord}_{x} (f^{- 1}) = - {ord}_{x} (f)$ , $div (f g) = div (f) + div (g)$ , principal divisors have degree zero, and linearly equivalent divisors have equal degree. (Divisors on a smooth proper curve, Order codimension one rational function, Degree divisor proper curve)

[F3]

$L (D) = {f \in k (C)^{\times} : div (f) + D \geq 0} \cup {0}$ is a $k$ -subspace of the function field, and the complete linear system $∣ D ∣ = {D^{'} effective : D^{'} \sim D}$ is in bijection with the set $P (L (D))$ of $k$ -lines in $L (D)$ via $f \mapsto div (f) + D$ ; when $L (D)$ is finite-dimensional, this is the $k$ -rational point set of its projective scheme of lines. In particular $∣ D ∣ = \emptyset$ exactly when $L (D) = 0$ . (The space L(D), Complete linear system)

[F4]

Under Choice, generating global sections $s_{0}, \dots, s_{n}$ of an invertible sheaf $L$ on an $S$ -scheme $X$ determine a unique $S$ -morphism $φ : X \to P_{S}^{n}$ with $φ^{*} O (1) ≅ L$ , $φ^{*} x_{i} = s_{i}$ , and chart formula $x_{j}^{(i)} \circ φ = s_{j} / s_{i}$ on the locus $X_{s_{i}}$ where $s_{i}$ is invertible; a closed point $x$ is a base point of a subspace $V \subseteq L (D)$ exactly when every nonzero $f \in V$ vanishes at $x$ , i.e. $x \in Supp (div (f) + D)$ for all such $f$ , and base-point-freeness is equivalent to the corresponding sections generating $O_{C} (D)$ . (Generating line-bundle sections define a morphism to projective space, Base points and base-point-free linear systems, Invertible sheaf of cartier divisor)

[F5]

A base-point-free subspace $V \subseteq L (D)$ of dimension $r + 1 \geq 1$ determines a $k$ -morphism $φ_{V} : C \to P_{k}^{r}$ with $φ_{V}^{*} O (1) ≅ O_{C} (D)$ under which the coordinate sections pull back to a basis of $V$ , and the members of $P (V)$ are exactly the pullbacks of hyperplanes. (A base-point-free linear system defines a morphism to projective space)

[F6]

For $d \geq 1$ the degree- $d$ Veronese map is $ν_{1, d} : P_{k}^{1} \to P_{k}^{d}$ , $[x_{0} : x_{1}] \mapsto [x_{0}^{d} : x_{0}^{d - 1} x_{1} : \dots : x_{1}^{d}]$ , and it is a well-defined closed immersion. (The degree-d Veronese map, The Veronese map is a well-defined closed immersion)

[F7]

Under Choice, two $S$ -morphisms from a reduced scheme to a separated $S$ -scheme agreeing on a dense open subscheme are equal. (Agreement on a schematically dense open)

[F8]

$k [t]$ is a principal ideal domain and a unique factorisation domain; every nonzero polynomial is a unit multiple of a product of monic irreducibles. (For every field $F$ , $F [x]$ is a principal ideal domain, Every principal ideal domain is a unique factorisation domain)

[F9]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

technique · direct; factor the divisors of polynomials, solve the effectivity inequalities for $L(d[\infty])$, and identify the associated morphism with the Veronese map on a dense chart

1.1F1F2F8

Divisors of polynomials. By [F1] a monic irreducible $g \in k [t]$ of degree $e$ has $div (g) = [V (g)] - e [\infty]$ , in particular $div (t) = [V (t)] - [\infty]$ ; factor a nonzero $p \in k [t]$ as $p = c \prod_{i} g_{i}^{e_{i}}$ with monic irreducibles $g_{i}$ and $c \in k^{\times}$ [F8]. Additivity of $ord$ and $div (f g) = div (f) + div (g)$ [F2] give $div (p) = \sum_{i} e_{i} [V (g_{i})] - (\sum_{i} e_{i} \deg g_{i}) [\infty] = Z (p) - (\deg p) [\infty],$ where $Z (p) = \sum_{i} e_{i} [V (g_{i})]$ is effective with $\deg_{k} Z (p) = \sum_{i} e_{i} \deg (g_{i}) = \deg p$ and is supported away from $\infty$ ; for a constant $p = c$ this reads $div (c) = 0$ .

2.1F1F2F3F8

The Riemann-Roch spaces of $d [\infty]$ . Let $f = P / Q \neq 0$ with coprime $P, Q \in k [t]$ [F8]; by step 1.1, $div (f) = Z (P) - Z (Q) + (\deg Q - \deg P) [\infty]$ , and $f \in L (d [\infty])$ means $div (f) + d [\infty] \geq 0$ [F3]. If $Q = c t^{m}$ then $div (f) + d [\infty] = Z (P) - m [0] + (m - \deg P + d) [\infty]$ , where $0 = V (t)$ is the origin and $t ∤ P$ by coprimality; effectivity forces $m = 0$ , so $f = P \in k [t]$ and $\deg P \leq d$ . If $Q$ has an irreducible factor $g \neq t$ , then $div (f)$ has the strictly negative coefficient $- e$ at $V (g) \neq \infty$ , so $div (f) + d [\infty]$ is not effective. Hence $L (d [\infty]) = {P \in k [t] : P = 0 or \deg P \leq d},$ the polynomials of degree at most $d$ together with $0$ : a $k$ -vector space with basis $1, t, \dots, t^{d}$ for $d \geq 0$ , of dimension $d + 1$ , and the zero space for $d < 0$ . Moreover $\deg_{k} (d [\infty]) = d \cdot [κ (\infty) : k] = d$ because $κ (\infty) = k$ [F2].

2.2F4F5F6F7F9step 1.1

The associated morphism is the Veronese embedding. Let $d \geq 1$ and let $V = L (d [\infty])$ , a $(d + 1)$ -dimensional subspace of $L (d [\infty])$ [F3]. No point of $P_{k}^{1}$ is a base point of $V$ : at a closed point $x \neq \infty$ the constant function $1$ does not vanish, since $div (1) + d [\infty] = d [\infty]$ is supported at $\infty$ , while at $x = \infty$ the polynomial $t^{d}$ does not vanish, since $div (t^{d}) + d [\infty] = d [0]$ is supported at the origin $0$ [F4, step 1.1]. So $V$ is base-point-free, and [F5] attaches to it a $k$ -morphism $φ_{V} : P_{k}^{1} \to P_{k}^{d}$ whose pullback of the coordinate sections is the basis $1, t, \dots, t^{d}$ of $V$ and whose hyperplane pullbacks are the members of $∣ d [\infty] ∣$ ; the same morphism is obtained from the generating sections $1, t, \dots, t^{d}$ of the invertible sheaf $O (d [\infty])$ by the universal property [F4]. On the chart $x_{0} \neq 0$ of $P_{k}^{d}$ the chart formula of [F4] gives $x_{j}^{(0)} \circ φ_{V} = t^{j}$ on the open where the section $1$ is invertible, which is $P_{k}^{1} ∖ {\infty}$ ; hence $φ_{V} ([1 : t]) = [1 : t : t^{2} : \dots : t^{d}]$ for every $t \neq \infty$ . The degree- $d$ Veronese map $ν_{1, d}$ of [F6] reads $[x_{0} : x_{1}] \mapsto [x_{0}^{d} : x_{0}^{d - 1} x_{1} : \dots : x_{1}^{d}] = [1 : t : \dots : t^{d}]$ on the same chart ${x_{0} \neq 0} = P_{k}^{1} ∖ {\infty}$ . Since $P_{k}^{1}$ is reduced and $P_{k}^{d}$ is separated over $k$ , [F7] gives $φ_{V} = ν_{1, d}$ ; by [F6] this morphism is a closed immersion, the degree- $d$ Veronese embedding.

3.1F1F2F3step 2.1

The complete linear system of $d [\infty]$ . For $d \geq 0$ let $D^{'}$ be an effective divisor of degree $d$ on $P_{k}^{1}$ ; by [F1] every divisor on $P_{k}^{1}$ is linearly equivalent to $\deg_{k} (D^{'}) [\infty] = d [\infty]$ , so $D^{'} \in ∣ d [\infty] ∣$ [F3]. Conversely every $D^{'} \in ∣ d [\infty] ∣$ is effective by definition and has $\deg_{k} D^{'} = \deg_{k} (d [\infty]) = d$ , since linearly equivalent divisors have equal degree [F2]. Hence $∣ d [\infty] ∣ = {D^{'} effective on P_{k}^{1} : \deg_{k} D^{'} = d}$ for $d \geq 0$ , a set parametrised by $P (L (d [\infty]))$ and hence by the projective space $P^{d}$ of $k$ -lines in the $(d + 1)$ -dimensional space $L (d [\infty])$ ; for $d < 0$ the space $L (d [\infty])$ is zero by step 2.1, so $∣ d [\infty] ∣ = \emptyset$ .

4.1F1F4F5F7F8F9step 1.1step 2.1step 3.1step 2.2∎

Conclusion. For every nonzero polynomial $p (t)$ of degree $m$ one has $div (p) = Z (p) - m [\infty]$ with $Z (p)$ effective of degree $m$ (step 1.1); the space $L (d [\infty])$ is exactly the space of polynomials of degree at most $d$ together with $0$ , of dimension $d + 1$ for $d \geq 0$ and zero for $d < 0$ (step 2.1); the complete linear system $∣ d [\infty] ∣$ is the set of effective divisors of degree $d$ for $d \geq 0$ and is empty for $d < 0$ (step 3.1). Its set of members is the set of $k$ -lines in $L (d [\infty])$ , identified for $d \geq 0$ with the $k$ -rational points of its projective scheme of lines. For $d \geq 1$ the associated morphism is the degree- $d$ Veronese closed immersion; for $d = 0$ it is the constant map to $P_{k}^{0}$ ; and for $d < 0$ there is no associated projective morphism. Choice enters through the current divisor, cohomology, Cartier/Weil and projective-space suppliers [F1], [F4], [F5], [F7], [F8] and [F9]; AC supplies DC for the Cartier-to-Weil interface. No further selection occurs.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-10-02Open item page →

A smooth conic is a projective line once it has a rational point

Example

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field of characteristic not two, let $C = V_{+} (F) \subseteq P_{k}^{2}$ be a smooth conic with a $k$ -rational point $p$ , and let $P_{k}^{1}$ be the projective line with its standard charts (Relative projective space from standard charts, Two-affine projective line and its twists). Then:

projection from $p$ exhibits $C$ as isomorphic to $P_{k}^{1}$ : the residual-intersection parametrisation $Φ : P_{k}^{1} \to C$ , $[X : Z] \mapsto [f X Z : - a X^{2} - e X Z - c Z^{2} : f Z^{2}]$ in the normal form of step 1.1, is an isomorphism of $k$ -schemes;
consequently, for the divisor $D = [P]$ of a $k$ -rational point $P \in C (k)$ , the Riemann-Roch space $L (D)$ (The space L(D)) has $k$ -dimension $2$ ;
the plane-curve arithmetic genus formula gives $p_{a} (C) = \frac{(2 - 1) (2 - 2)}{2} = 0$ (Arithmetic genus of a plane curve), so a smooth conic has genus zero; and since $C$ is smooth, hence normal, $C$ agrees with its normalization (Normalization of an integral finite-type curve by gluing affine integral closures).

Supplier interface. The projective-line calculation uses the earlier local lemma Projective-line curve and divisor basics. Its Proof 1.2 computes the residue degrees and Proof 2.1 computes the divisor of a monic irreducible polynomial; these are the statements used in [F4] and step 1.2.

Facts & Assumptions

Given: A field $k$ of characteristic not two, a nonzero homogeneous quadratic form $F \in k [X, Y, Z]$ , the conic $C = V_{+} (F) \subseteq P_{k}^{2}$ assumed smooth with $C (k) \neq \emptyset$ , a $k$ -rational point $p \in C (k)$ , and a $k$ -rational point $P \in C (k)$ .

[F1]

A curve over $k$ is geometrically integral, separated, of finite type and of chain dimension one; properness and smoothness are additional properties. (Curves over a field)

[F2]

Under Choice, $X \to Spec k$ is smooth if and only if for every field extension $K / k$ every local ring of the base change $X_{K}$ is regular; in particular smoothness implies regularity of the local rings of $X$ itself. Regular local rings are integrally closed domains, so an integral smooth scheme is normal. (Smoothness over a field by geometric regularity, regular local rings are normal)

[F3]

The projective plane $P_{k}^{2}$ and the projective line $P_{k}^{1}$ have their standard charts; $P_{k}^{1}$ has the charts $U_{0} = Spec k [t]$ and $U_{1} = Spec k [u]$ glued along $t u = 1$ , with $\infty = [0 : 1]$ the pole of $t$ , and the origin $[1 : 0] = V (t)$ the zero of $t$ . (Relative projective space from standard charts, Two-affine projective line and its twists)

[F4]

(Earlier local prerequisite.) On $P_{k}^{1}$ with coordinate $t$ : (1) $P_{k}^{1}$ is a smooth proper geometrically integral curve of genus $0$ ; (2) for every monic irreducible $g \in k [t]$ of degree $d$ , the closed point $p = V (g)$ has $[κ (p) : k] = d$ and $div (g) = [p] - d [\infty]$ . (Projective-line curve and divisor basics)

[F5]

On a smooth curve the order ${ord}_{x}$ of a nonzero rational function at a closed point is additive and satisfies ${ord}_{x} (f^{- 1}) = - {ord}_{x} (f)$ , the divisor of a rational function is $div (f) = \sum_{x} {ord}_{x} (f) [x]$ with $div (f g) = div (f) + div (g)$ , effectivity means all coefficients are nonnegative, and $\deg_{k} (\sum_{x} n_{x} [x]) = \sum_{x} n_{x} [κ (x) : k]$ . (Divisors on a smooth proper curve, Order codimension one rational function, Degree divisor proper curve)

[F6]

The Riemann-Roch space of a divisor $D$ on $C$ is $L (D) = {f \in k (C)^{\times} : div (f) + D \geq 0} \cup {0}$ , a $k$ -subspace of the function field. (The space L(D))

[F7]

Under Choice every rational map from a smooth curve to a proper $k$ -scheme is represented by a $k$ -morphism, and rational maps are equivalence classes of morphisms on nonempty opens. (Rational maps from a smooth curve to a proper scheme are morphisms, Rational maps of integral finite-type schemes)

[F8]

Under Choice, two $S$ -morphisms $a, b : W \to Y$ with $Y \to S$ separated and $W$ reduced agree if they agree on a dense open subscheme. (Agreement on a schematically dense open)

[F9]

Let $F$ be a nonzero homogeneous form of degree $d \geq 1$ with $X = V_{+} (F) \subseteq P_{k}^{2}$ an integral curve; then $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = 1 - χ (O_{X}) = \frac{(d - 1) (d - 2)}{2}$ . (Arithmetic genus of a plane curve)

[F10]

Under the Axiom of Choice, for an integral separated finite-type curve $C$ of chain dimension one the normalization $ν : C^{n u} \to C$ is integral and normal, finite and birational over $C$ , and initial among normal integral schemes finite and birational over $C$ . (Normalization of an integral finite-type curve by gluing affine integral closures, The Axiom of Choice)

[F11]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

[F12]

A finite-type $k$ -algebra is Noetherian; a finite-type domain $A$ over $k$ has $\dim A = {trdeg}_{k} Frac (A)$ , and the chain dimension of a Noetherian space is the supremum of dimensions on an open cover. (Finite-variable polynomial algebras over fields are Noetherian by finite generators, Affine-domain dimension equals transcendence degree, Dimension can be computed on an open cover)

[F13]

Projective space over $k$ is proper, a closed immersion is proper, and proper morphisms compose; a closed subscheme of $P_{k}^{2}$ is therefore proper over $k$ . (Finite-dimensional projective space is proper over every base, Closed immersions are proper, Properness survives composition)

Proof

technique · direct; put the conic into a normal form at the rational point, write down the residual-intersection parametrisation and the projection, glue them into an isomorphism on dense opens, and read off the Riemann-Roch space from the explicit model of the projective line

1.1F1F2F12F13

Geometric integrality, normal form, and scheme dimension. Since $C$ is smooth, [F2] says it remains regular after every field extension, in particular over an algebraic closure $\overset{ˉ}{k}$ . If $F_{\overset{ˉ}{k}}$ were reducible, a quadratic factorization would be either two distinct lines, singular at their intersection, or a repeated line, singular along that line; both contradict regularity. Thus $F_{\overset{ˉ}{k}}$ is irreducible and $C$ is geometrically integral. Choose homogeneous coordinates with $p = [0 : 1 : 0]$ and tangent line $T_{p} C = V (Z)$ . Writing $F = a X^{2} + b Y^{2} + c Z^{2} + d X Y + e X Z + f Y Z$ , the point condition gives $b = 0$ ; the tangent condition gives $d = 0$ and $f \neq 0$ , so $F = a X^{2} + e X Z + f Y Z + c Z^{2}$ . If $a = 0$ , then $F = Z (e X + f Y + c Z)$ , contradicting geometric integrality; hence $a \neq 0$ . The charts $D (Z)$ and $D (Y)$ cover $C$ , since the only projective point with $Y = Z = 0$ would be $[1 : 0 : 0]$ , where $F = a \neq 0$ . On $D (Z)$ , the equation $a x^{2} + e x + f y + c = 0$ is linear in $y$ with coefficient $f \neq 0$ , so the coordinate ring is $k [x]$ . On $D (Y)$ , the ring is the domain $R = k [x, z] / (a x^{2} + e x z + f z + c z^{2})$ . Its defining polynomial has positive degree in $z$ because $f \neq 0$ . The degree-in- $z$ product rule shows $k [x] ↪ R$ : a nonzero polynomial in $x$ cannot be a multiple of a polynomial of positive $z$ -degree. The equation also makes $z$ algebraic over $k (x)$ , so ${trdeg}_{k} Frac (R) = 1$ . Both finite-type chart rings are Noetherian by [F12]; the same fact makes $C$ Noetherian, and [F12] gives chart dimension one and chain dimension one for $C$ by the finite open-cover lemma. Thus [F1] makes $C$ a curve; it is proper by [F13] and smooth by assumption.

1.2F3F4F5

The model computation on the projective line. Let $P_{k}^{1}$ have coordinate $t$ on $U_{0}$ and point at infinity $\infty = [0 : 1]$ , the pole of $t = x_{1} / x_{0}$ [F3]. For a monic irreducible $g \in k [t]$ of degree $d$ , [F4] gives $div (g) = [V (g)] - d [\infty]$ ; if $P$ is a nonzero polynomial with factorization $P = c \prod_{i} g_{i}^{e_{i}}$ , then [F5] gives $div (P) = \sum_{i} e_{i} [V (g_{i})] - (\deg P) [\infty]$ . Let $Q \in P^{1} (k)$ . If $Q = \infty$ , write a nonzero $f \in L ([\infty])$ as $P / Q_{0}$ with coprime $P, Q_{0} \in k [t]$ . Any irreducible factor $g^{e}$ of a nonconstant denominator contributes coefficient $- e$ at the finite point $V (g)$ in $div (f) + [\infty]$ , since $P$ and $Q_{0}$ are coprime. Thus $Q_{0}$ is constant and $f = P$ ; then $div (f) + [\infty]$ is effective exactly when $\deg P \leq 1$ , so $L ([\infty]) = k \cdot 1 \oplus k \cdot t$ has dimension two. If $Q = [1 : c]$ for $c \in k$ , then $div (t - c) = [Q] - [\infty]$ [F4]. For $f = P / Q_{0}$ in lowest terms, every irreducible denominator factor other than $t - c$ would contribute a negative coefficient at its finite point, so $Q_{0} = (t - c)^{m}$ . Coprimeness gives $t - c ∤ P$ , and effectivity at $Q$ requires $1 - m \geq 0$ , hence $m \in {0, 1}$ . At infinity the coefficient is $m - \deg P$ , so $\deg P \leq m$ . If $m = 0$ , $P$ is constant; if $m = 1$ , write $P = α (t - c) + β$ , giving $f = α + β / (t - c)$ . Therefore $L ([Q]) = k \cdot 1 \oplus k \cdot \frac{1}{t - c}$ has dimension two.

1.3F1F2F9F10

Arithmetic genus and normalization. By [F1] the conic $C$ is an integral curve in $P_{k}^{2}$ , so [F9] gives $H^{0} (C, O_{C}) = k$ and $p_{a} (C) = \frac{(2 - 1) (2 - 2)}{2} = 0$ . By [F2] the local rings of the smooth curve $C$ are regular, hence integrally closed, so $C$ is normal; then the identity morphism $C \to C$ is a normal integral scheme, finite and birational over $C$ , so by initiality of the normalization [F10] the normalization $ν : C^{n u} \to C$ is an isomorphism, i.e. $C$ agrees with its normalization.

2.1F3step 1.1

The parametrization and the projection. Keep the normal form of step 1.1 and let $[X : Z]$ be homogeneous coordinates on $P_{k}^{1}$ . Define $Φ : P_{k}^{1} ⟶ P_{k}^{2}, [X : Z] ⟼ [f X Z : - (a X^{2} + e X Z + c Z^{2}) : f Z^{2}],$ whose components are homogeneous of degree two; they do not all vanish, because $Z = 0$ forces $X \neq 0$ and then the image is $[0 : - a X^{2} : 0] = [0 : 1 : 0]$ since $a \neq 0$ , so $Φ$ is a $k$ -morphism [F3]; and $Φ$ lands in $C$ : substituting gives $a (f X Z)^{2} + e (f X Z) (f Z^{2}) + f (- (a X^{2} + e X Z + c Z^{2})) (f Z^{2}) + c (f Z^{2})^{2} = f Z^{2} \cdot 0$ . In the other direction the projection $Π : C ∖ {p} ⟶ P_{k}^{1}, [X : Y : Z] ⟼ [X : Z],$ is a $k$ -morphism: on $C$ the equations $X = Z = 0$ define the single point $p = [0 : 1 : 0]$ , as $F (0, Y, 0) = 0$ for all $Y$ , so off $p$ at least one of $X, Z$ is nonzero.

3.1step 1.1step 2.1

The two maps are mutually inverse on dense opens. For $[X : Z] \in P_{k}^{1}$ with $Z \neq 0$ one has $Π (Φ ([X : Z])) = Π ([f X Z : - (a X^{2} + e X Z + c Z^{2}) : f Z^{2}]) = [f X Z : f Z^{2}] = [X : Z]$ , so $Π \circ Φ$ is the identity on the dense open ${Z \neq 0} \subseteq P_{k}^{1}$ . On the open chart $Z \neq 0$ of $C$ , with homogeneous coordinates $[X : Y : Z]$ , one has $Z \neq 0$ (if $Z = 0$ then $0 = F (X, Y, 0) = a X^{2}$ forces $X = 0$ , hence $[X : Y : Z] = p$ ), and the conic equation $Y f Z = - (a X^{2} + e X Z + c Z^{2})$ gives $Φ (Π ([X : Y : Z])) = [f X Z : - (a X^{2} + e X Z + c Z^{2}) : f Z^{2}] = [X (f Z) : Y (f Z) : Z (f Z)] = [X : Y : Z]$ , since $f Z \neq 0$ ; so $Φ \circ Π$ is the identity on the dense open $C ∖ {p}$ .

4.1F7F8F11step 2.1step 3.1

The isomorphism. The morphism $Π$ of step 2.1 represents a rational map $C ⇢ P_{k}^{1}$ [F7]; the curve $C$ is smooth and $P_{k}^{1}$ is proper over $k$ , so under Choice [F11] the extension lemma [F7] represents this rational map by a morphism $\overline{Π} : C \to P_{k}^{1}$ extending $Π$ . The morphisms $Φ \circ \overline{Π}$ and ${i d}_{C}$ from the reduced scheme $C$ to the separated $k$ -scheme $C$ agree on the dense open $C ∖ {p}$ by step 3.1, so they are equal by [F8]; similarly $\overline{Π} \circ Φ$ and ${i d}_{P^{1}}$ agree on the dense open ${Z \neq 0} \subseteq P^{1}$ by step 3.1, so $\overline{Π} \circ Φ = {i d}_{P^{1}}$ . Hence $\overline{Π}$ is an isomorphism of $k$ -schemes with inverse $Φ$ , which is the first assertion: projection from $p$ exhibits $C ≅ P_{k}^{1}$ .

5.1F5F6step 1.2step 4.1

The Riemann-Roch space of a rational point. Let $D = [P]$ with $P \in C (k)$ and put $Q = \overline{Π} (P) \in P_{k}^{1} (k)$ . An isomorphism of $k$ -schemes induces a $k$ -isomorphism of function fields and a bijection of closed points preserving residue fields, hence a degree-preserving bijection of divisor groups intertwining $div$ and $ord$ by [F5]; under the isomorphism $\overline{Π} : C \to P_{k}^{1}$ the pullback of $[Q]$ is $[P] = D$ , and pullback of rational functions $f \mapsto f \circ \overline{Π}$ carries $L ([Q])$ onto $L (D)$ [F6]. By step 1.2 the space $L ([Q])$ is $2$ -dimensional over $k$ , with basis ${1, t}$ for $Q = \infty$ and ${1, \frac{1}{t - c}}$ for the finite point $Q = [1 : c]$ with $t = x_{1} / x_{0}$ ; hence $\dim_{k} L (D) = 2$ .

6.1F2F4F7F8F10step 1.3step 4.1step 5.1∎

Conclusion. For a smooth conic $C \subseteq P_{k}^{2}$ with a $k$ -rational point $p$ , step 4.1 exhibits an explicit isomorphism $C ≅ P_{k}^{1}$ from the projection at $p$ and its residual-intersection parametrization, step 5.1 computes $\dim_{k} L ([P]) = 2$ for every $k$ -rational point $P$ , and step 1.3 gives $p_{a} (C) = \frac{(2 - 1) (2 - 2)}{2} = 0$ together with the agreement of $C$ with its normalization. The Axiom of Choice is assumed for the geometric-regularity characterization and normalization in steps 1.1 and 1.3, as well as the rational-map extension and dense-open uniqueness in step 4.1 [F2, F10, F7, F8]. The divisor calculation in step 1.2 uses the verified clauses 1 and 2 of the current draft supplier [F4].

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-10-02Open item page →

Ramification of the double cover y^2=f(x)

Example

Assume the Axiom of Choice (The Axiom of Choice) for the current normalization, curve/function-field, finiteness, smooth-differential, properness, ampleness, and Čech-cohomology supplier routes used below. Let $k$ be algebraically closed of characteristic $\neq 2$ and let $f \in k [x]$ be squarefree of degree $n = 2 g + 1$ or $n = 2 g + 2$ with $g \geq 1$ . The affine curve $y^{2} = f (x)$ has a smooth projective model $C$ , and the projection to the $x$ -line is a finite surjective morphism $π : C \to P_{k}^{1}$ of degree two that is branched exactly at the roots of $f$ and, when $n$ is odd, at infinity: over each of these $2 g + 2$ branch points there lies exactly one point with $e_{p} = 2$ , and over every other closed point of $P_{k}^{1}$ there lie two points with $e_{p} = 1$ . The plane model $X$ of $C$ has degree $n$ and $p_{a} (X) = \frac{(n - 1) (n - 2)}{2}$ . Its only possible singular point is infinity; that point is singular exactly when $n \geq 4$ , with $δ_{\infty} = \frac{(n - 1) (n - 2)}{2} - g$ . For $n = 3$ (so $g = 1$ ), the plane cubic is smooth and the singularity sum is empty, with $δ_{\infty} = 0$ . In all cases $g (C) = g$ .

The ramification assertions use only $char k \neq 2$ and squarefreeness of $f$ . The genus computation is carried out for algebraically closed $k$ , because the delta invariant (Delta invariant of a curve singularity) and the plane-curve genus correction (Geometric genus of a plane curve by delta invariants) are stated over algebraically closed fields.

Facts & Assumptions

Given: An algebraically closed field $k$ with $char k \neq 2$ , a squarefree polynomial $f \in k [x]$ of degree $n = 2 g + 1$ or $n = 2 g + 2$ , $g \geq 1$ , the affine curve $U = V (y^{2} - f) \subseteq A_{k}^{2}$ , the plane model $X = V_{+} (G) \subseteq P_{k}^{2}$ with $G = Y^{2} Z^{n - 2} - F (X, Z)$ and $F (X, Z) = Z^{n} f (X / Z)$ , and the normalization $ν : C \to X$ ; the Axiom of Choice is assumed.

[F1]

Jacobian criterion over an algebraically closed field: a closed point of $V (h) \subseteq A_{k}^{2}$ is regular exactly when the two partial derivatives of $h$ do not both vanish there. (Jacobian rank detects regularity at closed points)

[F2]

For an integral plane curve $X = V_{+} (F)$ of degree $d$ one has $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = \frac{(d - 1) (d - 2)}{2}$ . (Arithmetic genus of a plane curve)

[F3]

The normalization $ν : C \to X$ is finite, birational, $C$ is integral and normal with $k (C) = k (X)$ , and over the algebraically closed field $k$ the curve $C$ is smooth; for a smooth proper geometrically connected curve the genus is $g (C) = h^{1} (C, O_{C}) = 1 - χ (O_{C})$ , and the geometric genus of $X$ is $g (X) = g (C)$ . (Normalization of an integral finite-type curve by gluing affine integral closures, Geometric genus of a singular curve, Genus and arithmetic genus of a curve, Curves over a field)

[F4]

The delta invariant is $δ_{x} (X) = \dim_{k} ((ν_{*} O_{C})_{x} / O_{X, x})$ , it vanishes exactly at regular points, and $p_{a} (X) = g (X^{n u}) + \sum_{x} δ_{x} (X)$ , equivalently $g (X^{n u}) = \frac{(d - 1) (d - 2)}{2} - \sum_{x} δ_{x} (X)$ for the plane model of degree $d$ . (Delta invariant of a curve singularity, Arithmetic genus, geometric genus and delta invariants, Geometric genus of a plane curve by delta invariants)

[F5]

Dominant $k$ -morphisms $C \to D$ of smooth proper geometrically integral curves correspond bijectively to injective $k$ -algebra homomorphisms $k (D) ↪ k (C)$ ; a nonconstant morphism is finite, surjective and has degree $\deg (f) = [k (C) : k (D)]$ . (Smooth proper curves, dominant morphisms and function fields, Degree of a nonconstant morphism of curves, Nonconstant morphisms of proper curves are finite and surjective)

[F6]

The ramification index is $e_{p} = {ord}_{p}$ of the pullback of a uniformizer of the target; for every closed point $q$ one has $\sum_{p \in f^{- 1} (q)} e_{p} [κ (p) : κ (q)] = \deg (f)$ ; the differential-ramification locus consists of the points with $e_{p} > 1$ together with those having inseparable residue extension, and over the algebraically closed field $k$ the residue extensions are trivial. (Ramification index of a morphism of curves, Fibre degree sum with ramification and residue degrees, Ramification points, branch points and unramifiedness)

[F7]

At a closed point $p$ of the smooth curve $C$ the local ring $O_{C, p}$ is a discrete valuation ring, ${ord}_{p}$ is a valuation with ${ord}_{p} (u v) = {ord}_{p} (u) + {ord}_{p} (v)$ , and an element is a uniformizer exactly when its order is one. (Local rings at closed points of smooth curves are discrete valuation rings, Every nonzero fraction is a unit times a power of a uniformiser)

[F8]

The canonical bundle is $ω_{C} = Ω_{C / k}^{1}$ and is locally free of rank one (Canonical bundle and canonical divisors, Differentials of a smooth morphism). At a closed point $p$ of $C$ , the residue field $κ (p)$ is finite over $k$ by the finite-type residue-field lemma, hence equals $k$ because $k$ is algebraically closed (Finite-type maps from Jacobson rings induce finite residue-field extensions at maximal ideals). The cotangent sequence for this finite separable residue extension identifies $m_{p} / m_{p}^{2}$ with $Ω_{C / k}^{1} \otimes κ (p)$ (Separable residue and the cotangent sequence of a local algebra). The local ring is a discrete valuation ring, so a uniformizer $t_{p}$ gives a basis of the one-dimensional space $m_{p} / m_{p}^{2}$ (Local rings at closed points of smooth curves are discrete valuation rings); consequently $d t_{p}$ gives a basis of $Ω_{C / k}^{1} \otimes κ (p)$ and Nakayama's lemma (Assuming the Axiom of Choice, Nakayama's lemma) makes $d t_{p}$ a local frame of $ω_{C}$ near $p$ . Thus for a nonzero rational differential $ω = g d t_{p}$ its order is ${ord}_{p} (g)$ , independently of the chosen uniformizer, agreeing with the frame definition (Canonical bundle and canonical divisors). Also $div (ω) = \sum_{p} {ord}_{p} (ω) [p]$ is effective exactly when $ω$ is a global section of $ω_{C}$ , two nonzero rational differentials on a smooth proper geometrically integral curve differ by a nonzero rational function $g$ , and $div (g ω) = div (ω) + div (g)$ (Canonical bundle and canonical divisors).

[F9]

The two-affine double-cover calculation gives $\dim_{k} H^{1} (C, O_{C}) = g$ when the separated scheme has affine rings $k [x, y] / (y^{2} - f)$ and $k [t, w] / (w^{2} - ψ)$ , intersection obtained by inverting $x$ or $t$ , and transition $t = x^{- 1}$ , $w = x^{- (g + 1)} y$ . (Cohomology of a two-chart double cover)

[F10]

Projectivity of the model: for every scheme $S$ the structure morphism $P_{S}^{m} \to S$ is proper; a morphism factoring as a closed immersion into $P_{k}^{m}$ followed by the projection is proper, so the plane model $X$ is proper over $k$ ; the twisting sheaf $O (1)$ on $P_{k}^{1}$ is ample and for every finite morphism $g : Y \to P_{k}^{1}$ the pullback $g^{*} O (1)$ is ample; a finite morphism is proper, a composite of a finite morphism with a proper morphism is proper, and a composite of finite-type morphisms is of finite type; finally, if $Z$ is proper of finite type over a Noetherian scheme $S$ and $L$ is ample on $Z$ , then for every sufficiently large $d$ the power $L^{\otimes d}$ is closed H-very ample relative to $S$ , so that some $S$ -closed immersion $Z ↪ P_{S}^{N}$ pulls $O (1)$ back to $L^{\otimes d}$ . (Finite-dimensional projective space is proper over every base, Projective morphisms are proper, The projective-line twisting sheaf is ample, Finite pullback preserves absolute ampleness, Finite morphisms are proper, Composite of a finite morphism and a proper morphism is proper, High powers of an ample line bundle embed a proper scheme)

[F11]

A regular Noetherian ring is normal, and a normal domain is integrally closed in its fraction field: every element of the fraction field integral over the ring lies in it. In particular the local rings of a smooth affine curve over $k$ are regular, so the curve and all its local rings are integrally closed. (regular local rings are normal, normal noetherian ring)

Proof technique: direct; identify the plane model and its only possible singular point, compute the ramification of the degree-two projection from the local normal forms, compute the genus of the smooth model by writing every regular differential against the explicit canonical divisor $d x / y$ , and finish with the plane-curve delta correction.

Verification

1.1F1

The affine curve is smooth. Write $h = y^{2} - f (x) \in k [x, y]$ ; its partial derivatives are $2 y$ and $- f^{'} (x)$ . At a common zero one has $y = 0$ (characteristic $\neq 2$ ) and $f (x) = f^{'} (x) = 0$ , so $x$ is a multiple root of $f$ , contrary to squarefreeness. By the Jacobian criterion every closed point of $U$ is regular.

1.2F2

The plane model. The polynomial $G$ is homogeneous of degree $n$ and $Z ∤ G$ , because $G (X, Y, 0) = - F (X, 0) = - a_{n} X^{n}$ with $a_{n} \neq 0$ ; its dehomogenization at $Z = 1$ is $h = y^{2} - f (x)$ , which is irreducible in $k [x, y]$ since $\deg_{y} h = 2$ and $f$ is squarefree of degree at least three, hence not a square in $k [x]$ . If $G = G_{1} G_{2}$ were a factorization into nonconstant forms, substituting $Z = 1$ would express the irreducible $h$ as a product, so one factor would dehomogenize to a nonzero constant $c$ ; that factor is homogeneous of positive degree and equals $c$ on the hyperplane $Z = 1$ at that point, hence is the form $c Z^{\deg G_{i}}$ , forcing $Z ∣ G$ , a contradiction. Therefore $G$ is irreducible, $X$ is an integral plane curve of degree $n$ , and $U = X \cap {Z \neq 0}$ is a dense open subscheme. By the plane-curve formula $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = \frac{(n - 1) (n - 2)}{2}$ .

2.1F1step 1.1

The only possible singular point. On the chart $Y \neq 0$ put $u = X / Y$ , $v = Z / Y$ , so that $X$ is cut out by $g (u, v) = v^{n - 2} - F (u, v)$ , where $F (u, v) = \sum_{i = 0}^{n} a_{i} u^{i} v^{n - i}$ is a binary form of degree $n$ with $a_{n} \neq 0$ . The point $\infty = (0 : 1 : 0)$ corresponds to $(u, v) = (0, 0)$ , and every term of $F$ and of its first partial derivatives has order at least $n - 1 \geq 2$ at the origin; hence the partials of $g$ at the origin are $- F_{u} = 0$ and $(n - 2) v^{n - 3} - F_{v}$ , which vanish at the origin for $n \geq 4$ and equal $(0, 1)$ for $n = 3$ . Every other closed point of the chart lies either in $U$ , which is regular by step 1.1, or has $Z = 0$ ; but $Z = 0$ on $X$ forces $X = 0$ , so the only such point is $\infty$ itself. Hence $\infty$ is the only possibly singular point of $X$ , and it is singular exactly when $n \geq 4$ .

2.2F5step 1.2

A degree-two projection. The function field of $X$ is $k (X) = k (x, y)$ with $y^{2} = f (x)$ (step 1.2), and $y^{2} - f$ is irreducible over $k (x)$ because $f$ is squarefree of degree at least three and hence is not a square in $k (x)$ ; therefore $[k (X) : k (x)] = 2$ . By [F5] the inclusion $k (x) ↪ k (C) = k (X)$ determines a dominant morphism $π : C \to P_{k}^{1}$ with $π^{*} (x) = x$ , and $π$ is finite and surjective of degree two.

3.1F1F11step 1.1step 2.2

The two affine charts. Finiteness of $π$ (step 2.2) makes the preimages $U^{'} = π^{- 1} (Spec k [x])$ and $C_{t} = π^{- 1} (Spec k [t])$ of the two standard affine charts of $P_{k}^{1}$ affine, with coordinate rings $A^{'}$ and $B^{'}$ module-finite over $k [x]$ and over $k [t]$ respectively (Finite morphisms of schemes), and $Frac A^{'} = Frac B^{'} = k (C)$ . Put $t = 1 / x$ , $w = y t^{g + 1}$ , and $ψ (t) = t^{n} f (1 / t)$ when $n = 2 g + 2$ , $ψ (t) = t^{n + 1} f (1 / t)$ when $n = 2 g + 1$ ; then $w^{2} = ψ (t)$ in $k (C)$ . The subrings $A = k [x, y] / (y^{2} - f) \subseteq A^{'}$ and $B = k [t, w] / (w^{2} - ψ) \subseteq B^{'}$ have $Frac A = k (x, y) = k (C)$ and $Frac B = k (t, w) = k (C)$ , the latter because $[k (C) : k (t)] = [k (C) : k (x)] = 2$ by step 2.2 while $w \notin k (t)$ . The polynomial $ψ$ is squarefree: its roots are the inverses of the nonzero roots of $f$ , all simple because $f$ is squarefree, together with $t = 0$ in the odd case, where $ψ = t φ$ with $φ (t) = t^{n} f (1 / t)$ of constant term $a_{n} \neq 0$ , so that root is simple as well. Since $char k \neq 2$ , the Jacobian criterion [F1] shows that the localizations of $A$ (step 1.1) and of $B$ at maximal ideals are regular local rings, their localizations at the zero ideal are fraction fields, and so $A$ and $B$ are regular Noetherian domains, hence integrally closed [F11]; the same applies to $A^{'}$ and $B^{'}$ , which are coordinate rings of affine opens of the smooth curve $C$ . Now $x \in A^{'}$ and $y^{2} = f (x) \in A^{'}$ with $y \in k (C) = Frac A^{'}$ , so $y \in A^{'}$ and $A \subseteq A^{'}$ ; dually $t \in B^{'}$ and $w^{2} = ψ (t) \in B^{'}$ give $w \in B^{'}$ and $B \subseteq B^{'}$ . Every element of $A^{'}$ is integral over $k [x] \subseteq A$ and every element of $B^{'}$ is integral over $k [t] \subseteq B$ , so integrally closedness gives $A^{'} = A$ and $B^{'} = B$ . Consequently the closed points of the finite chart are the maximal ideals of $A$ , the points of $C$ over $\infty$ are the maximal ideals of $B$ lying over $(t)$ , and the local rings of $C$ at these points are the corresponding localizations.

4.1F6F7step 3.1

Ramification over finite points. Fix $x_{0} \in k$ and let $p \in C$ lie over $x_{0}$ , so that $p$ corresponds to a maximal ideal of $A$ (step 3.1), while the pullback of the uniformizer $x - x_{0}$ of $P_{k}^{1}$ at $x_{0}$ is the function $x - x_{0}$ , whence $e_{p} = {ord}_{p} (x - x_{0})$ . If $f (x_{0}) \neq 0$ , choose $y_{0} \in k^{\times}$ with $y_{0}^{2} = f (x_{0})$ and write $f (x) - f (x_{0}) = (x - x_{0}) u (x)$ , without requiring $u (x_{0}) \neq 0$ . At $p = (x_{0}, y_{0})$ the factor $y + y_{0}$ is a unit, so $(y - y_{0}) = (x - x_{0}) u (x) / (y + y_{0})$ in the local ring and its maximal ideal $(x - x_{0}, y - y_{0})$ is $(x - x_{0})$ . At $p^{'} = (x_{0}, - y_{0})$ the factor $y - y_{0}$ is a unit, and the same identity gives $(y + y_{0}) = (x - x_{0}) u (x) / (y - y_{0})$ , so the maximal ideal $(x - x_{0}, y + y_{0})$ is again $(x - x_{0})$ . Thus $x - x_{0}$ is a uniformizer at both points and $e_{p} = 1$ , regardless of whether $x_{0}$ is a critical point of $f$ ; these are exactly the two points over $x_{0}$ . If $f (x_{0}) = 0$ , write $f = (x - x_{0}) u (x)$ with $u (x_{0}) \neq 0$ ; then $x - x_{0} = y^{2} u (x)^{- 1}$ in $k (C)$ , so $x - x_{0}$ lies in the square of the maximal ideal at the unique point $p = (x_{0}, 0)$ of $U$ and generates its square, whence the maximal ideal is generated by $y$ , making $y$ a uniformizer with ${ord}_{p} (x - x_{0}) = 2$ , $e_{p} = 2$ , and the fibre is ${p}$ . In both cases the fibre-degree formula of [F6] shows that no further point lies over $x_{0}$ , because the displayed contributions already sum to $\deg π = 2$ with trivial residue extensions.

4.2F6F7step 3.1

Ramification over infinity. The points of $C$ over $\infty$ are the maximal ideals of $B$ over $(t)$ , that is, the maximal ideals of $B / (t) = k [w] / (w^{2} - ψ (0))$ (step 3.1). If $n$ is even, then $ψ (0) = a_{n} \neq 0$ and $k [w] / (w^{2} - a_{n}) ≅ k \times k$ , so there are exactly two points $p_{\pm}$ over $\infty$ , and $w$ is a unit at each of them. At $p_{+}$ with $c = \sqrt{a_{n}}$ , the identity $(w - c) (w + c) = ψ (t) - ψ (0) = t g (t)$ , where $g = (ψ - ψ (0)) / t \in k [t]$ , shows that $w - c = t g (t) (w + c)^{- 1} \in (t)$ because $w + c$ has nonzero residue $2 c$ and is a unit; hence the maximal ideal $(t, w - c)$ equals $(t)$ and $t$ is a uniformizer at $p_{+}$ , and the same computation with $- c$ at $p_{-}$ gives a uniformizer $t$ there too. If $n$ is odd, then $ψ (0) = 0$ and $B / (t) = k [w] / (w^{2})$ is local with maximal ideal $(w)$ , so there is exactly one point $p_{\infty}$ over $\infty$ ; since $ψ = t φ$ with $φ (0) = a_{n} \neq 0$ , the element $φ (t)$ is a unit at $p_{\infty}$ , so $t = w^{2} φ (t)^{- 1} \in (w)^{2}$ . In the local ring $B_{(t, w)}$ , one has $B_{(t, w)} / (w) ≅ k [t]_{(t)} / (t φ (t)) ≅ k$ , so its maximal ideal $(t, w)$ is $(w)$ ; hence $w$ is a uniformizer at $p_{\infty}$ and ${ord}_{p_{\infty}} (t) = 2$ . In both cases the pullback of the uniformizer $t$ of $P_{k}^{1}$ at $\infty$ is the function $t$ itself, so the orders just computed are the ramification indices: $e_{p_{+}} = e_{p_{-}} = 1$ for even $n$ , and $e_{p_{\infty}} = 2$ for odd $n$ .

4.3F3F9F10step 2.2step 3.1

Genus and projectivity. The proper plane model $X$ and finite normalization make $C$ proper of finite type by [F3, F10]. The finite projection gives the ample sheaf $L = π^{*} O (1)$ by [F10]. A positive power of $L$ is closed H-very ample over the Noetherian base field by [F10], so $C$ is projective. It is smooth of dimension one by [F3]. The two affine charts in step 3.1 are the inverse images of the standard cover of $P_{k}^{1}$ ; their intersection is obtained by inverting $x$ or $t$ , with $t = x^{- 1}$ and $w = x^{- (g + 1)} y$ . Properness supplies separatedness. Thus [F9] applies and gives $h^{1} (C, O_{C}) = g$ , with classes $x^{- 1} y, \dots, x^{- g} y$ . The genus definition in [F3] now gives $g (C) = g$ .

5.1F6step 4.1step 4.2

The branch locus. Combining step 4.1 and step 4.2: over a point $x_{0} \in k$ with $f (x_{0}) \neq 0$ there are exactly two points of $C$ , both with $e = 1$ ; over each of the $n$ roots of $f$ there is exactly one point, with $e = 2$ ; and over $\infty$ there are two points with $e = 1$ when $n$ is even and one point with $e = 2$ when $n$ is odd. Since $k$ is algebraically closed, every residue field extension at these closed points is trivial, so the fibre-degree formula $\sum_{p \in π^{- 1} (q)} e_{p} [κ (p) : κ (q)] = \deg π = 2$ holds at every closed point $q$ of $P_{k}^{1}$ , consistently with the counts. The ramification locus of $π$ is therefore the set of $n + [n odd] = 2 g + 2$ points over the roots of $f$ and, for odd $n$ , over infinity, all of index two, and the branch locus is exactly the set of the $n$ roots of $f$ together with $\infty$ when $n$ is odd, again $2 g + 2$ points; over each branch point lies exactly one ramification point, and over every other closed point of $P_{k}^{1}$ lie two points with $e = 1$ .

5.2F7F8step 4.1step 4.2

The differential $ω_{0} = d x / y$ and its divisor. The form $ω_{0} = d x / y$ is a nonzero rational differential on $C$ [F8]: $y \neq 0$ in the field $k (C)$ , and $d x \neq 0$ because $k (C) / k (x)$ is separable of degree two in characteristic $\neq 2$ . At a point over $x_{0}$ with $f (x_{0}) \neq 0$ , step 4.1 provides the uniformizer $x - x_{0}$ and $y$ is a unit, so $ord (d x / y) = ord (y^{- 1} d (x - x_{0})) = 0$ ; at the point over a root $x_{0}$ of $f$ , writing $f = (x - x_{0}) u$ with $u (x_{0}) \neq 0$ gives $x - x_{0} = y^{2} u (x)^{- 1}$ and therefore $d x / y = 2 u (x)^{- 1} (1 + y^{2} u (x)^{- 2} u^{'} (x))^{- 1} d y .$ The coefficient is a unit at that point, so $d x / y$ is a unit multiple of $d y$ and has order $0$ ; this formula shows that $d x$ itself is not a unit multiple of $d y$ there. Over infinity, $x = 1 / t$ , $d x = - t^{- 2} d t$ and $y = w t^{- (g + 1)}$ give $d x / y = - t^{g - 1} d t / w$ . In the even case $t$ is a uniformizer and $w$ a unit at $p_{\pm}$ (step 4.2), so ${ord}_{p_{\pm}} (d x / y) = g - 1$ ; in the odd case $t = w^{2} φ (t)^{- 1}$ gives $d t = 2 w φ (t)^{- 1} (1 + w^{2} φ (t)^{- 2} φ^{'} (t))^{- 1} d w$ , so that $d t / w$ is a unit multiple of $d w$ , and $t^{g - 1} = w^{2 g - 2} φ (t)^{- (g - 1)}$ gives ${ord}_{p_{\infty}} (d x / y) = 2 g - 2$ . Hence $div (d x / y) = (g - 1) (p_{+} + p_{-})$ in the even case and $div (d x / y) = (2 g - 2) p_{\infty}$ in the odd case; both divisors have degree $2 g - 2$ , and each is a canonical divisor on the smooth proper geometrically integral curve $C$ .

5.3F4F7step 1.2step 2.1step 4.3

The delta invariant at infinity. By [F4] one has $p_{a} (X) = g (X^{n u}) + \sum_{x} δ_{x} (X)$ , the sum over the closed points of $X$ ; since $δ_{x}$ vanishes at regular points and $\infty$ is the only possibly singular point of $X$ (step 2.1), the sum reduces to $δ_{\infty}$ , and it is empty when $n = 3$ , where $\infty$ is regular and $δ_{\infty} = 0$ . With $p_{a} (X) = \frac{(n - 1) (n - 2)}{2}$ (step 1.2) and $g (X^{n u}) = g (C) = g$ (step 4.3), $δ_{\infty} = \frac{(n - 1) (n - 2)}{2} - g,$ which is $2 g^{2}$ for $n = 2 g + 2$ , $2 g (g - 1)$ for $n = 2 g + 1$ , and $0$ for $(n, g) = (3, 1)$ ; in particular it is a nonnegative integer in each family.

6.1F8step 2.2step 3.1step 5.2

Regular differentials. Let $ω$ be a global section of the canonical bundle $ω_{C} = Ω_{C / k}^{1}$ , that is, a regular differential on $C$ [F8]; the zero section is the case $R = 0$ below. If $ω \neq 0$ , then $ω = h ω_{0}$ for a unique $h \in k (C)^{\times}$ [F8]. Since $ω_{0}$ has order $0$ at every point of the finite chart and generates the free rank-one module $ω_{C, p}$ there (step 5.2), regularity of $ω$ forces $h \in O_{C} (U^{'}) = A = k [x, y] / (y^{2} - f)$ (step 3.1), so $h = R (x) + S (x) y$ for unique $R, S \in k [x]$ , the elements $1, y$ forming a $k (x)$ -basis of $k (C)$ (step 2.2, step 3.1). Regularity on $U^{'}$ is then automatic, and by [F8] it remains to impose at the points over infinity the condition ${ord}_{p} (h) \geq - {ord}_{p} (ω_{0})$ coming from $div (ω) = div (h) + div (ω_{0})$ . In the even case, at $p_{\pm}$ one has $ord (R (x)) = - \deg R$ and $ord (S (x) y) = - (g + 1) - \deg S$ for nonzero $R$ and $S$ , using $y = w t^{- (g + 1)}$ with $w$ a unit, while ${ord}_{p_{\pm}} (ω_{0}) = g - 1$ ; hence ${ord}_{p_{\pm}} (h) \geq - (g - 1)$ is required. For $S = 0$ this is exactly $\deg R \leq g - 1$ . For $S \neq 0$ : if the two orders differ, then $ord (h)$ is the smaller one, and since $- (g + 1) - \deg S \leq - (g + 1) < - (g - 1)$ this is impossible; if the two orders are equal, so that $\deg R = g + 1 + \deg S$ , then the coefficient of $t^{- \deg R}$ in $h$ at $p_{\pm}$ is $r + w (p_{\pm}) s$ with leading coefficients $r, s \neq 0$ , and regularity at both $p_{+}$ and $p_{-}$ would force $r + \sqrt{a_{n}} s = r - \sqrt{a_{n}} s = 0$ , impossible in characteristic $\neq 2$ with $a_{n} \neq 0$ . Hence $S = 0$ and $\deg R \leq g - 1$ . In the odd case, at $p_{\infty}$ one has $ord (R (x)) = - 2 \deg R$ and $ord (S (x) y) = - (2 g + 1) - 2 \deg S$ for nonzero $R$ and $S$ , while ${ord}_{p_{\infty}} (ω_{0}) = 2 g - 2$ ; if $S \neq 0$ , then $- (2 g + 1) - 2 \deg S \leq - 2 g - 1 < - (2 g - 2)$ , so if the orders of the two summands differ the order of $h$ is too small, and they cannot be equal because $- 2 \deg R$ is even while $- (2 g + 1) - 2 \deg S$ is odd. Hence again $S = 0$ , and $- 2 \deg R \geq - (2 g - 2)$ gives $\deg R \leq g - 1$ . Conversely, every $R \in k [x]$ with $\deg R \leq g - 1$ yields a regular differential $R (x) d x / y$ : it is regular on $U^{'}$ , and at infinity its order is $g - 1 - \deg R \geq 0$ in the even case and $2 (g - 1 - \deg R) \geq 0$ in the odd case (step 5.2, step 4.2). Therefore $H^{0} (C, ω_{C}) = {R (x) d x / y : R \in k [x], \deg R \leq g - 1},$ the differentials $d x / y, x d x / y, \dots, x^{g - 1} d x / y$ are linearly independent over $k$ , and $h^{0} (C, ω_{C}) = g$ .

7.1F3F5F6F8F9F10step 1.1step 1.2step 2.1step 2.2step 3.1step 4.1step 4.2step 5.1step 5.2step 6.1step 4.3step 5.3∎

Conclusion. Step 1.1 shows that the affine curve $y^{2} = f (x)$ is smooth, and step 1.2 and step 2.1 identify the plane model $X = V_{+} (G)$ as an integral plane curve of degree $n$ with $p_{a} (X) = \frac{(n - 1) (n - 2)}{2}$ whose only possibly singular point is $\infty$ , singular exactly when $n \geq 4$ . Step 2.2 exhibits the degree-two projection $π : C \to P_{k}^{1}$ from the normalization, step 3.1 identifies the two standard affine charts with the explicit rings $A = k [x, y] / (y^{2} - f)$ and $B = k [t, w] / (w^{2} - ψ)$ , and step 4.1 and step 4.2 compute the ramification over the finite points and over infinity. Step 5.1 shows that $π$ is finite and surjective of degree two, branched exactly at the $n$ roots of $f$ and, when $n$ is odd, at infinity — that is, at $2 g + 2$ branch points — with exactly one ramification point of index two over each of them and two points with $e = 1$ over every other closed point of $P_{k}^{1}$ . Step 5.2 and step 6.1 compute $div (d x / y)$ and identify $H^{0} (C, ω_{C})$ with the $g$ -dimensional space of differentials $R (x) d x / y$ with $\deg R \leq g - 1$ , and step 4.3 computes $H^{1} (C, O_{C})$ from the two affine charts and concludes $g (C) = g$ . Finally step 5.3 computes the delta invariant $δ_{\infty} = \frac{(n - 1) (n - 2)}{2} - g$ , equal to $2 g^{2}$ for $n = 2 g + 2$ , to $2 g (g - 1)$ for $n = 2 g + 1$ and to $0$ for $(n, g) = (3, 1)$ , so that the geometric genus of $X$ is $g$ . The Axiom of Choice is used through the normalization and curve/function-field/finiteness routes [F3], [F5], [F6], the differential and Čech-cohomology routes [F8], [F9], and the properness and ampleness suppliers [F10], at the steps where those inputs are applied.

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A torsion-only extension of the canonical formula fails for Frobenius

Statement refuted

The proposed extension $ω_{C} ≅ f^{*} ω_{D} \otimes O_{C} (R_{f}^{t o r})$ of the canonical bundle formula to every finite surjective morphism of smooth proper geometrically integral curves is false when $R_{f}^{t o r} = \sum_{p} {length}_{O_{C, p}} (tors (Ω_{C / D})_{p}) [p],$ where the summation is over closed points and $tors$ means the torsion subsheaf of the relative differentials. The separable theorem Canonical bundle formula with the different defines its different divisor only when $k (C) / k (D)$ is separable; $R_{f}^{t o r}$ here is a proposed candidate extension, not that theorem's different divisor. The witness is the $p$ -th-power map in characteristic $p > 0$ , $φ ⁣ : P_{k}^{1} \to P_{k}^{1}$ , $[s : t] \mapsto [s^{p} : t^{p}]$ . Its relative differentials are invertible, so their torsion subsheaf is zero and $R_{f}^{t o r} = 0$ . But $ω_{P_{k}^{1}} ≅ O (- 2 [\infty])$ and $φ^{*} ω_{P_{k}^{1}} ≅ O (- 2 p [\infty])$ are not isomorphic. This refutes extending the separable formula by the torsion-submodule recipe; it does not assign the separable different divisor to an inseparable map. The indices at $0$ and $\infty$ are both $p$ .

Facts & Assumptions

Given: A field $k$ of characteristic $p > 0$ , the projective line with coordinates $x$ on $U_{0}$ and $y = x^{- 1}$ on $U_{\infty}$ , and the morphism $φ$ given by $x \mapsto x^{p}$ and $y \mapsto y^{p}$ . The Axiom of Choice is assumed wherever required by the cited projective-line, finite-map, and principal-divisor suppliers below.

[F1]

The projective line has the two affine charts $U_{0} = Spec k [x]$ and $U_{\infty} = Spec k [y]$ , with $x y = 1$ on their overlap; it is a smooth proper geometrically integral curve, and its closed-point local rings are discrete valuation rings. The point $\infty$ has uniformizer $y$ . (Two-affine projective line and its twists, Relative projective space from standard charts, Curves over a field, Local rings at closed points of smooth curves are discrete valuation rings)

[F2]

A nonconstant rational function on a smooth proper geometrically integral curve defines a finite locally free map to $P^{1}$ of degree the corresponding function-field extension; for $x^{p}$ , $[k (x) : k (x^{p})] = p$ , with basis $1, x, \dots, x^{p - 1}$ . The fibre degree formula is $\sum_{r \mapsto q} e_{r} [κ (r) : κ (q)] = \deg (φ)$ . (A nonconstant rational function defines a finite map to the projective line, Fibre degree sum with ramification and residue degrees, The Axiom of Choice)

[F3]

At a closed point on a smooth curve the local ring is a discrete valuation ring. The ramification index is the order of the pullback of a target uniformizer; a uniformizer has order one, and orders are additive. (Local rings at closed points of smooth curves are discrete valuation rings, Every nonzero fraction is a unit times a power of a uniformiser, Ramification index of a morphism of curves)

[F4]

For $B = A [T] / (g (T))$ , the relative differentials are generated by $d T$ with relation $g^{'} (T) d T = 0$ . (Differentials of a polynomial quotient and the Jacobian cokernel)

[F5]

For smooth curves the canonical sheaf is $ω = Ω_{- / k}^{1}$ and is invertible. The different divisor in The different divisor of a generically separable morphism of curves is defined from the lengths of the full relative-differential stalks only when the function-field extension is separable. (Canonical bundle and canonical divisors, Sheaf of relative Kähler differentials, The different divisor of a generically separable morphism of curves, Canonical bundle formula with the different)

[F6]

For ring maps $A \to B \to C$ , the Kähler differential sequence $C \otimes_{B} Ω_{B / A} ⟶ Ω_{C / A} ⟶ Ω_{C / B} ⟶ 0$ is exact; its first map need not be injective. This applies without separability. (Transitivity sequence for differentials)

[F7]

On $P_{k}^{1}$ , $d x$ is a rational section of the canonical line bundle and its divisor is $- 2 [\infty]$ , as follows from $d x = - y^{- 2} d y$ . The rational-section and Cartier-divisor dictionary identifies a line bundle with the sheaf of a divisor of any nonzero rational section; for the finite flat map $φ$ , pullback of a Cartier divisor computes the pullback of its line bundle. An isomorphism of divisor line bundles makes their difference principal. Principal divisors on a proper curve have degree zero, and $\deg_{k} (\sum n_{q} [q]) = \sum n_{q} [κ (q) : k]$ . (Canonical bundle and canonical divisors, Rational sections of line bundles are Cartier divisors, Invertible sheaf of cartier divisor, Pullback of a Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group, Degree divisor proper curve, Principal divisors on a normal proper curve have degree zero)

Construction

Let $k$ be a field of characteristic $p > 0$ and let $φ ⁣ : P_{k}^{1} \to P_{k}^{1}$ be the morphism with $φ^{♯} (x) = x^{p}$ on the standard chart, that is $φ ([s : t]) = [s^{p} : t^{p}]$ in homogeneous coordinates. Its degree is $p$ , and $e_{0} = e_{\infty} = p$ . After base change to an algebraic closure, every geometric closed point is index-ramified with index $p$ ; the two displayed points are not the only geometric ramification points.

Verification

1.1F1F2F3

The morphism, its degree and its ramification. The coordinate function $x$ has ${ord}_{\infty} (x) = - 1$ by [F1], so $x^{p}$ is nonconstant, and [F2] gives the finite surjective morphism $φ$ of degree $[k (P_{k}^{1}) : k (x^{p})] = p$ , since $1, x, \dots, x^{p - 1}$ is a basis over $k (x^{p})$ . On the two charts the maps are $k [x] \to k [x]$ , $x \mapsto x^{p}$ , and $k [y] \to k [y]$ , $y \mapsto y^{p}$ , because $y = x^{- 1}$ . The points $0$ and $\infty$ are $k$ -rational with uniformizers $x$ and $y$ by [F1], so [F3] gives $e_{0} = {ord}_{0} (x^{p}) = p$ and $e_{\infty} = {ord}_{\infty} (y^{p}) = p$ . The zero and pole fibres are supported respectively at $0$ and $\infty$ ; the fibre degree formula [F2] at $0$ is consistent with $\deg (φ) = e_{0} [κ (0) : κ (0)] = p$ .

1.2F3algebra

Ramification after geometric base change. Over $\overset{ˉ}{k}$ , let $a$ be any finite target point and choose $b \in \overset{ˉ}{k}$ with $b^{p} = a$ . The pullback of the target parameter $x - a$ is $x^{p} - a = (x - b)^{p}$ , so the index at the geometric point $b$ is $p$ . On the infinity chart the same calculation is $y \mapsto y^{p}$ . Thus every geometric closed point is index-ramified. Over an imperfect original field the indices of its closed points need not all be $p$ : for example, if $k = F_{p} (a)$ with $a$ not a $p$ -th power, the target point $t = a$ has preimage defined by the irreducible polynomial $x^{p} - a$ . Its local uniformizer is $x^{p} - a$ , exactly the pullback of $t - a$ , so its index is $1$ , while the residue extension is purely inseparable.

2.1F5F6step 1.1

The pullback of differentials is the zero map. The canonical bundle of the target is generated on $U_{0}$ by $d x$ , and the pullback map $φ^{*} ω_{P_{k}^{1}} \to ω_{P_{k}^{1}}$ sends its generator $φ^{*} (d x)$ to $d (x^{p}) = p x^{p - 1} d x = 0$ , because $p = 0$ in $k$ . The same computation in coordinate $y$ on $U_{\infty}$ gives $φ^{*} (d y) \mapsto d (y^{p}) = 0$ . By the right-exact transitivity sequence [F6], the map $φ^{*} ω_{P_{k}^{1}} \to ω_{P_{k}^{1}}$ is followed by the quotient to $Ω_{C / D}$ . Since the first map is zero, this quotient is an isomorphism. Thus the pullback of differentials is not injective; the injectivity assertion in the canonical-bundle theorem [F5] is unavailable because its separability hypothesis fails.

2.2F4step 1.1

The relative differentials are invertible, with no torsion. On the source chart $U_{0}$ the map is $A = k [x] \to B = k [z]$ , $x \mapsto z^{p}$ , so $B = A [T] / (T^{p} - x)$ with $T \mapsto z$ ; the derivative of $g (T) = T^{p} - x$ is $g^{'} (T) = p T^{p - 1} = 0$ , and [F4] gives $Ω_{B / A} ≅ B / (g^{'}) = B$ , free of rank one on $d T$ . The same holds on $U_{\infty}$ in coordinate $y$ . Thus the relative differential sheaf for $φ$ , $Ω_{C / D}$ , is invertible, so its torsion subsheaf is zero. Set $l_{p}^{t o r} : = {length}_{O_{C, p}} (tors (Ω_{C / D})_{p})$ ; then $l_{p}^{t o r} = 0$ for every closed point $p$ .

3.1F5F7step 1.1step 2.2

The recipe gives $R_{f}^{t o r} = 0$ , but the isomorphism fails. By step 2.2 all coefficients $l_{p}^{t o r}$ vanish. The canonical divisor computation is $div (d x) = - 2 [\infty]$ : $d x$ is a frame on $U_{0}$ , and $d x = d (y^{- 1}) = - y^{- 2} d y$ on $U_{\infty}$ . Thus $ω_{P_{k}^{1}} ≅ O_{P_{k}^{1}} (- 2 [\infty])$ . The pullback divisor is $φ^{*} [\infty] = p [\infty]$ , since its fibre is supported at $\infty$ with index $p$ from step 1.1; [F7] gives $φ^{*} ω_{P_{k}^{1}} ≅ O_{P_{k}^{1}} (- 2 p [\infty])$ . If the proposed formula held with $R_{f}^{t o r} = 0$ , these divisor line bundles would be isomorphic, so $(2 p - 2) [\infty]$ would be principal by [F7]. Its degree is $2 p - 2 > 0$ , contradicting [F7], which gives degree zero for principal divisors.

4.1F1F5F6F7step 1.1step 1.2step 2.1step 2.2step 3.1∎

Conclusion. The $p$ -th-power map is finite surjective of degree $p$ ; its indices at $0$ and $\infty$ are $p$ , and after base change every geometric closed point has index $p$ . Its relative differentials are invertible with zero torsion, so the proposed torsion-submodule recipe gives $R_{f}^{t o r} = 0$ . The formula fails because $ω_{P_{k}^{1}} ≅ O (- 2 [\infty])$ and $φ^{*} ω_{P_{k}^{1}} ≅ O (- 2 p [\infty])$ are not isomorphic. Thus separability cannot be dropped when extending the formula by this recipe, and this conclusion does not define the separable different divisor for an inseparable map.

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Nodal cubic: arithmetic genus one, delta one, geometric genus zero

Example

Assume the Axiom of Choice. Let $k$ be algebraically closed of characteristic $\neq 2$ and let $X = V_{+} (Y^{2} Z - X^{3} - X^{2} Z) \subseteq P_{k}^{2}$ be the nodal plane cubic, with node $o = (0 : 0 : 1)$ . Then $p_{a} (X) = \frac{(3 - 1) (3 - 2)}{2} = 1$ , the unique singular point $o$ is a node with $δ_{o} (X) = 1$ , and the normalization $X^{n u}$ has geometric genus $g (X^{n u}) = 1 - 1 = 0$ ; the normalization is the projective line, matching the parametrization $(T : S) \mapsto (S (T^{2} - S^{2}) : T (T^{2} - S^{2}) : S^{3})$ of the nodal cubic. (Characteristic two is excluded because there the tangent cone degenerates and the singular point is not an ordinary node; the computation below uses $char k \neq 2$ .)

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ with $char k \neq 2$ , the plane cubic $X = V_{+} (Y^{2} Z - X^{3} - X^{2} Z)$ , its node $o = (0 : 0 : 1)$ , and the map $ψ : P_{k}^{1} \to X$ , $(T : S) \mapsto (S (T^{2} - S^{2}) : T (T^{2} - S^{2}) : S^{3})$ .

[F1]

For an integral proper plane curve $X = V_{+} (F) \subseteq P_{k}^{2}$ with $d = \deg F$ one has $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = \frac{(d - 1) (d - 2)}{2}$ . (Arithmetic genus of a plane curve)

[F2]

Under the Axiom of Choice, for an integral proper finite-type curve over algebraically closed $k$ with normalization $ν$ , the delta invariant $δ_{x} (X) = \dim_{k} ((ν_{*} O_{X^{n u}})_{x} / O_{X, x})$ vanishes exactly at regular points, and $g (X^{n u}) = p_{a} (X) - \sum_{x} δ_{x} (X)$ ; the sum is finite and supported on the singular points, where it is computed. (Delta invariant of a curve singularity, Arithmetic genus, geometric genus and delta invariants, The Axiom of Choice)

[F3]

Under the Axiom of Choice, if $F$ is irreducible of degree $d$ defining the integral plane curve $X$ , then $g (X^{n u}) = \frac{(d - 1) (d - 2)}{2} - \sum_{x} δ_{x} (X)$ over the finitely many singular points. (Geometric genus of a plane curve by delta invariants, The Axiom of Choice)

[F4]

Under the Axiom of Choice, the normalization glues affine integral closures and is finite, birational and initial among normal integral schemes finite and birational over an integral separated finite-type curve of chain dimension one. (Normalization of an integral finite-type curve by gluing affine integral closures, The Axiom of Choice)

[F5]

At a closed $k$ -rational point of $k [x, y] / (f)$ , regularity is equivalent to Jacobian rank $2 - \dim A_{m}$ ; for the closed points of these integral curve charts the local dimension is one, so this is equivalent to the gradient of $f$ being nonzero. (Jacobian rank detects regularity at closed points)

[F6]

$P_{k}^{1}$ has its standard affine charts and is smooth, proper and geometrically integral; under Choice, regular local rings are normal, so $P_{k}^{1}$ is normal. (Two-affine projective line and its twists, regular local rings are normal, The Axiom of Choice)

[F7]

A finite-type $k$ -algebra is Noetherian; a finite-type domain $A$ over $k$ has $\dim A = {trdeg}_{k} Frac (A)$ , and the chain dimension of a Noetherian space is the supremum of the dimensions on an open cover. (Finite-variable polynomial algebras over fields are Noetherian by finite generators, Affine-domain dimension equals transcendence degree, Dimension can be computed on an open cover)

[F8]

Projective space over $k$ is proper, a closed immersion is proper, and proper morphisms compose; hence a closed subscheme of $P_{k}^{2}$ is proper over $k$ . (Finite-dimensional projective space is proper over every base, Closed immersions are proper, Properness survives composition)

[F9]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

technique · direct; locate the unique singular point by the Jacobian criterion, compute the delta invariant from the explicit normalization, and apply the genus correction formula

1.1F5F7F8

The cubic is an integral curve, and its only singular point is the ordinary node $o$ . On $Z = 1$ , let $f = y^{2} - x^{3} - x^{2} = y^{2} - x^{2} (x + 1)$ . Over $k (x)$ the monic quadratic $f$ is irreducible: $x^{2} (x + 1)$ has valuation one at $x + 1$ , so it is not a square; Gauss's lemma gives irreducibility in $k [x, y]$ . The homogeneous cubic is not divisible by $Z$ , hence is irreducible as well. Since $k$ is algebraically closed, $X$ is geometrically integral. The charts $D (Z)$ and $D (Y)$ cover $X$ , because $Y = Z = 0$ in its equation forces $X = 0$ . On $D (Z)$ the coordinate ring is the domain $A = k [x, y] / (y^{2} - x^{3} - x^{2})$ with fraction field $k (t)$ under $x = t^{2} - 1$ , $y = t (t^{2} - 1)$ ; here $t = y / x$ in the fraction field. On $D (Y)$ the coordinate ring is $R = k [x, z] / ((1 - x^{2}) z - x^{3})$ . The identity $1 = (1 - x^{2}) (1 + x^{2} + x z)$ in $R$ makes $1 - x^{2}$ invertible, and solving for $z$ gives $R ≅ k [x, (1 - x^{2})^{- 1}]$ . Both chart rings are finite-type domains and hence Noetherian; [F7] gives their dimension one and the chain dimension one of their finite open cover $X$ . The projective cubic is a closed subscheme of $P_{k}^{2}$ , hence proper by [F8]; it is also separated and finite type. Thus it is an integral proper curve, and its generic point is regular because its local ring is the function field. Every other point is closed. On $Z = 1$ the partials of $f$ are $- 3 x^{2} - 2 x = - x (3 x + 2)$ and $2 y$ . A closed singular point must have $y = 0$ , so the curve equation gives $x = 0$ or $x = - 1$ ; at $x = - 1$ the first partial is $- 1$ , in every characteristic. Thus the only singular point on this chart is $o$ . On $Y = 1$ the equation $g = z - x^{3} - x^{2} z$ has partials $- 3 x^{2} - 2 x z$ and $1 - x^{2}$ ; if the second vanishes, then $x^{2} = 1$ and $g = - x^{3} \neq 0$ , so there is no singular point there. On $X = 1$ the equation $h = y^{2} z - 1 - z$ has partials $2 y z$ and $y^{2} - 1$ ; the equation $z (y^{2} - 1) = 1$ forces the second partial to be nonzero. The quadratic tangent cone at $o$ is $y^{2} - x^{2} = (y - x) (y + x)$ , with distinct tangent lines because $char k \neq 2$ . Hence $o$ is an ordinary node and all other points are regular.

2.1F1step 1.1

Arithmetic genus. By step 1.1, $X$ is an integral proper plane curve of degree three, so [F1] gives $p_{a} (X) = \frac{(3 - 1) (3 - 2)}{2} = 1$ .

2.2F4F6step 1.1

Explicit normalization and finite projective map. Put $q = T^{2} - S^{2}$ . The homogeneous triple defining $ψ (T : S) = (S q : T q : S^{3})$ has no common zero: if $S = 0$ , then $Y = T^{3} \neq 0$ , while if $S \neq 0$ , then $Z = S^{3} \neq 0$ . It lands on $X$ because $Y^{2} Z = T^{2} q^{2} S^{3} = S^{3} q^{2} (q + S^{2}) = X^{3} + X^{2} Z$ . On $D (Z)$ the source chart is $Spec k [t]$ with $t = T / S$ , and its ring map is $A = k [x, y] / (y^{2} - x^{3} - x^{2}) \to k [t]$ , $x \mapsto t^{2} - 1$ , $y \mapsto t (t^{2} - 1)$ . It is finite because $k [t] = A [t]$ and $t^{2} = x + 1$ ; it is birational since $t = y / x$ in $Frac (A)$ . On $D (Y)$ the inverse image is $Spec k [u, (1 - u^{2})^{- 1}]$ , where $u = S / T$ , and the chart map $R \to k [u, (1 - u^{2})^{- 1}]$ sends $x \mapsto u$ , $z \mapsto u^{3} / (1 - u^{2})$ ; it is an isomorphism by the description of $R$ in step 1.1. As $D (Z)$ and $D (Y)$ cover $X$ , these chart maps show that $ψ$ is finite. Its source $P_{k}^{1}$ is normal and integral by [F6], so the finite birational map identifies it with the normalization by the initial property in [F4].

3.1F2step 1.1step 2.2

Delta at the node. By step 2.2, $ψ$ is the normalization map. The two points $t = 1$ and $t = - 1$ of the affine source map to $o$ and no other point does, so the stalk of $ν_{*} O_{P^{1}}$ at $o$ is the semilocalization $k [t]_{S}$ with $S = {h \in k [t] : h (1) \neq 0, h (- 1) \neq 0}$ . Put $u = t^{2} - 1$ and identify the coordinate ring of $Z = 1$ with the subring $A = k [u, t u] = k [u] \oplus t u k [u] \subseteq k [t]$ ; also $k [t] = k [u] \oplus t k [u]$ . The semilocalization equals $k [u]_{(u)} \oplus t k [u]_{(u)}$ : if $h (t) = a (u) + t b (u)$ is nonzero at both $1$ and $- 1$ , its norm $h (t) h (- t) = a (u)^{2} - (u + 1) b (u)^{2}$ is nonzero at $u = 0$ , so $h$ is invertible after localizing over $k [u]_{(u)}$ , and every element of $k [u] ∖ (u)$ is nonzero at both points. The image of $O_{X, o} = A_{(u, t u)}$ in this semilocalization is $k [u]_{(u)} \oplus t u k [u]_{(u)}$ . The quotient is $t k [u]_{(u)} / t u k [u]_{(u)} ≅ k [u]_{(u)} / u k [u]_{(u)} ≅ k$ , of dimension one. Hence $δ_{o} (X) = 1$ , and since $o$ is the only singular point by step 1.1 and delta vanishes off the singular locus, $\sum_{x} δ_{x} (X) = 1$ .

4.1F2F3step 3.1

Genus. By [F3] and steps 1.1 and 3.1, $g (X^{n u}) = \frac{(3 - 1) (3 - 2)}{2} - \sum_{x} δ_{x} (X) = 1 - 1 = 0$ ; the geometric genus of $X$ is $0$ .

5.1F4step 2.2step 4.1

Normalization is the line. By step 2.2, the displayed map is a normal integral finite birational model of $X$ ; the uniqueness clause in [F4] identifies it with $X^{n u}$ . Thus the normalization is the projective line, and the geometric genus computed in step 4.1 is zero.

6.1F4F6F2F3F9step 2.2step 3.1step 4.1step 5.1∎

Conclusion. Under the Axiom of Choice, for the nodal cubic $X = V_{+} (Y^{2} Z - X^{3} - X^{2} Z)$ over an algebraically closed field with $char k \neq 2$ , the arithmetic genus is $1$ , the unique singular point is the ordinary node $o$ with delta invariant $1$ , and the normalization is the projective line; the normalization has geometric genus $0$ . Choice is used in the normalization and normality interfaces [F4, F6] and the delta/genus interfaces [F2, F3], in steps 2.2, 3.1, 4.1 and 5.1.

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Cuspidal cubic: delta invariant and normalization

Example

Assume the Axiom of Choice. Let $k$ be algebraically closed with $char k \neq 2, 3$ and let $X = V_{+} (Y^{2} Z - X^{3}) \subseteq P_{k}^{2}$ be the cuspidal cubic, with cusp $o = (0 : 0 : 1)$ . Then $p_{a} (X) = \frac{(3 - 1) (3 - 2)}{2} = 1$ , the cusp has $δ_{o} (X) = 1$ , the curve is smooth away from the cusp, and the normalization of $X$ is the projective line via the parametrization $t ⟼ (t^{2} : t^{3} : 1),$ so the geometric genus of $X$ is $g (X) = 0$ .

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ of characteristic $\neq 2, 3$ , the plane curve $X = V_{+} (Y^{2} Z - X^{3})$ , its cusp $o = (0 : 0 : 1)$ , and the map $φ : P_{k}^{1} \to X$ given on coordinates $(T : S)$ by $(T : S) \mapsto (T^{2} S : T^{3} : S^{3})$ .

[F1]

For a closed subscheme $V_{+} (F) \subseteq P_{k}^{2}$ which is a curve, $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = \frac{(d - 1) (d - 2)}{2}$ , where $d = \deg F$ . (Arithmetic genus of a plane curve)

[F2]

Under the Axiom of Choice, for an integral proper finite-type curve $X$ over algebraically closed $k$ with normalization $ν : X^{n u} \to X$ , the delta invariant is $δ_{x} (X) = \dim_{k} ((ν_{*} O_{X^{n u}})_{x} / O_{X, x})$ , it vanishes exactly at regular points, the sum over closed points is finite and supported on the singular locus, and $g (X^{n u}) = p_{a} (X) - \sum_{x} δ_{x} (X)$ ; the geometric genus $g (X) = g (X^{n u})$ is the genus of the smooth proper normalization. (Delta invariant of a curve singularity, Arithmetic genus, geometric genus and delta invariants, Geometric genus of a singular curve, The Axiom of Choice)

[F3]

Under the Axiom of Choice, for an irreducible homogeneous form $F$ of degree $d \geq 1$ defining the integral plane curve $X$ , one has $g (X^{n u}) = \frac{(d - 1) (d - 2)}{2} - \sum_{x} δ_{x} (X)$ , the sum over the finitely many singular points. (Geometric genus of a plane curve by delta invariants, The Axiom of Choice)

[F4]

Under the Axiom of Choice, the normalization of an integral separated finite-type curve of chain dimension one glues the affine integral closures; it is finite and birational, unique up to unique isomorphism, and initial among normal integral schemes finite and birational over the curve. (Normalization of an integral finite-type curve by gluing affine integral closures, The Axiom of Choice)

[F5]

At a closed $k$ -rational point of an affine hypersurface $A = k [t_{1}, \dots, t_{n}] / (f)$ , regularity is equivalent to Jacobian rank $n - \dim A_{m}$ ; for the closed points of these integral curve charts the local dimension is one, so in two variables this is equivalent to the gradient of $f$ being nonzero. (Jacobian rank detects regularity at closed points)

[F6]

[F7]

[F8]

[F9]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

technique · direct; identify the singular locus by the Jacobian criterion, compute the delta invariant from the explicit normalization, and read the genus off the correction formula

1.1F5F7F8

The cubic is an integral curve, and its only singular point is the cusp $o$ . On $Z = 1$ , the equation is $f = y^{2} - x^{3}$ . Over $k (x)$ this monic quadratic is irreducible because $x^{3}$ has odd valuation at $x = 0$ and is not a square; Gauss's lemma gives irreducibility in $k [x, y]$ . The homogeneous cubic is not divisible by $Z$ , hence is irreducible. Since $k$ is algebraically closed, $X$ is geometrically integral. The charts $D (Z)$ and $D (Y)$ cover $X$ , because $Y = Z = 0$ in its equation forces $X = 0$ . On $D (Z)$ the coordinate ring $A = k [x, y] / (y^{2} - x^{3})$ is a domain with fraction field $k (t)$ under $x = t^{2}$ , $y = t^{3}$ , where $t = y / x$ . On $D (Y)$ the ring is $k [x, z] / (z - x^{3}) ≅ k [x]$ . Both chart rings are finite-type domains and hence Noetherian; [F7] gives their dimension one and the chain dimension one of their finite open cover $X$ . The projective cubic is a closed subscheme of $P_{k}^{2}$ , hence proper by [F8]; it is also separated and finite type. Thus it is an integral proper curve, and its generic point is regular because its local ring is the function field. Every other point is closed. On $Z = 1$ , the partials of $f$ are $- 3 x^{2}$ and $2 y$ , so a closed singular point must be $x = y = 0$ because $char k \neq 2, 3$ ; this is $o$ . On $Y = 1$ , the equation is $g = z - x^{3}$ and its partials $- 3 x^{2}$ and $1$ never vanish together. On $X = 1$ , the equation is $h = y^{2} z - 1$ with partials $2 y z$ and $y^{2}$ ; the equation forces $y \neq 0$ , so they do not both vanish. Hence all other points are regular.

2.1F1step 1.1

Arithmetic genus. By step 1.1, $X$ is an integral proper plane curve of degree three, so [F1] gives $p_{a} (X) = \frac{(3 - 1) (3 - 2)}{2} = 1$ .

2.2F4F6step 1.1

The finite projective normalization map. The homogeneous triple defining $φ (T : S) = (T^{2} S : T^{3} : S^{3})$ has no common zero: if $S = 0$ , then $Y = T^{3} \neq 0$ , while if $S \neq 0$ , then $Z = S^{3} \neq 0$ . It lands on $X$ because $Y^{2} Z = T^{6} S^{3} = X^{3}$ . On $D (Z)$ the source is $Spec k [t]$ with $t = T / S$ , and the ring map $A = k [x, y] / (y^{2} - x^{3}) \to k [t]$ , $x \mapsto t^{2}$ , $y \mapsto t^{3}$ , is finite because $k [t] = A [t]$ and $t^{2} = x$ ; it is birational since $t = y / x$ in $Frac (A)$ . On $D (Y)$ the inverse image is $Spec k [s]$ with $s = S / T$ , and the target ring $k [x, z] / (z - x^{3})$ maps isomorphically to $k [s]$ by $x \mapsto s$ , $z \mapsto s^{3}$ . As $D (Z), D (Y)$ cover $X$ , the projective map is finite. Its source is normal and integral by [F6], so the finite birational map identifies it with the normalization by the initial property in [F4].

3.1F2step 1.1step 2.2

Delta at the cusp. By step 2.2, $φ$ is the normalization map. The only point of the affine source mapping to the cusp $o$ is $t = 0$ , so the stalk of $ν_{*} O_{P^{1}}$ at $o$ is the local ring $k [t]_{(t)}$ , and the image of $O_{X, o} = A_{(x, y)}$ in it is the local ring $A_{(x, y)} = k [t^{2}, t^{3}]_{(t^{2}, t^{3})}$ . Writing $u = t^{2}$ one has $k [t] = k [u] \oplus t k [u]$ and $k [t^{2}, t^{3}] = k [u] \oplus t^{3} k [u] = k [u] \oplus t u k [u]$ ; localizing at $(t)$ , which inverts no power of $u$ , gives $k [t]_{(t)} = k [u]_{(u)} \oplus t k [u]_{(u)}$ and $k [t^{2}, t^{3}]_{(t^{2}, t^{3})} = k [u]_{(u)} \oplus t u k [u]_{(u)}$ . Hence the quotient is $t k [u]_{(u)} / t u k [u]_{(u)} ≅ k [u]_{(u)} / u k [u]_{(u)} ≅ k$ , a one-dimensional $k$ -vector space, and $δ_{o} (X) = 1$ . Since $o$ is the only singular point by step 1.1 and $δ$ vanishes at regular points, $\sum_{x} δ_{x} (X) = 1$ .

4.1

Genus. By [F3] applied to the cubic $X$ (which has only isolated singularities by step 1.1) and steps 2.1 and 3.1, [F2, F3, step 2.1, step 3.1] $g (X^{n u}) = \frac{(3 - 1) (3 - 2)}{2} - δ_{o} (X) = 1 - 1 = 0,$ so the geometric genus of $X$ is $0$ .

5.1F4step 2.2step 4.1

The normalization is the line. By step 2.2, the displayed map is a normal integral finite birational model of $X$ ; the uniqueness clause in [F4] identifies it with $X^{n u}$ . Hence the normalization of the cuspidal cubic is the projective line, and its geometric genus is zero by step 4.1.

6.1F4F6F2F3F9step 2.2step 3.1step 4.1step 5.1∎

Conclusion. Under the Axiom of Choice, for the cuspidal cubic $X = V_{+} (Y^{2} Z - X^{3})$ over an algebraically closed field with $char k \neq 2, 3$ : $p_{a} (X) = 1$ , the cusp $o$ is the unique singular point with $δ_{o} (X) = 1$ , and the normalization is $P_{k}^{1}$ via $t \mapsto (t^{2} : t^{3} : 1)$ with geometric genus zero. Choice is used in the normalization and normality interfaces [F4, F6] and the delta/genus interfaces [F2, F3], in steps 2.2, 3.1, 4.1 and 5.1.

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Smoothness of the source cannot be dropped in the extension of rational maps

Statement refuted

The claim that the smoothness hypothesis on the source can be weakened in the extension theorem for rational maps: for a curve $X$ that is integral but not smooth and a proper target, not every rational map $X ⇢ Y$ extends to a morphism. Concretely, on the nodal plane cubic the rational map given by the slope of the branch is defined on the smooth locus, its two branches carry two different boundary values at the node, and no morphism from the whole curve extends it. When an extension does exist on such a curve it is unique (source reduced, target separated), so the failure is existence, not uniqueness.

Facts & Assumptions

Given: A field $k$ of characteristic different from $2$ , the nodal cubic $X = Spec k [x, y] / (y^{2} - x^{2} (x + 1))$ , the morphism $ν : A^{1} = Spec k [t] \to X$ with $x \mapsto t^{2} - 1$ , $y \mapsto t (t^{2} - 1)$ , and the projective line $P_{k}^{1}$ with chart $U_{0} = Spec k [\hat{t}]$ . We work under the Axiom of Choice [A1].

[A1]

The Axiom of Choice is assumed (The Axiom of Choice).

[A2]

In ZF, AC implies Dependent Choice (AC implies DC implies countable choice). This supplies the DC use in the curve closed-subset finiteness route used by the smooth-curve extension theorem.

[F1]

A curve over $k$ is nonempty, geometrically integral, separated, finite type, and of chain dimension one; an affine scheme is integral exactly when its coordinate ring is a domain. (Curves over a field, Integral schemes)

[F2]

A rational map of integral finite-type $k$ -schemes into a separated finite-type $k$ -scheme is represented by a morphism on a nonempty open subscheme. (Rational maps of integral finite-type schemes)

[F3]

If $W$ is reduced, $V \subseteq W$ is a dense open subscheme and $Y \to S$ is separated, then two $S$ -morphisms $W \to Y$ agreeing on $V$ are equal. (Agreement on a schematically dense open)

[F4]

The projective line is glued from the charts $U_{0} = Spec k [\hat{t}]$ and $U_{\infty} = Spec k [\hat{u}]$ ; the chart coordinate defines a morphism to $P^{1}$ . (Two-affine projective line and its twists)

[F5]

$P_{S}^{n} \to S$ is proper, and every proper morphism is separated; in particular $P_{k}^{1}$ is separated over $k$ . (Finite-dimensional projective space is proper over every base)

[F6]

Under AC, every rational map from a smooth curve to a proper $k$ -scheme extends to a morphism. Its proof uses DC through the finiteness of proper closed subsets of a curve. (Rational maps from a smooth curve to a proper scheme are morphisms, Proper closed subsets of a curve are finite)

[F7]

A finite-variable polynomial ring over a field is a UFD and each irreducible is prime. For a UFD $R$ , a primitive polynomial in $R [y]$ is irreducible when it is irreducible over $Frac (R) [y]$ ; polynomial degrees add over a domain. (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Gauss lemma over a UFD, Over an integral domain, degrees add under multiplication of nonzero polynomials, $R / P$ is an integral domain if and only if $P$ is a prime ideal)

[F8]

A finite-type domain over $k$ has Krull dimension equal to the transcendence degree of its fraction field. The prime-spectrum correspondence identifies this with the chain dimension of its Noetherian spectrum; for a closed point, the affine local-dimension formula identifies the local dimension with the local-ring dimension when its residue field is algebraic over $k$ . (Affine-domain dimension equals transcendence degree, Finite-variable polynomial algebras over fields are Noetherian by finite generators, The spectrum of a Noetherian ring is a Noetherian topological space, A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point, Krull dimension of a nonzero ring, Chain dimension and the empty-space convention, Local fibre dimension equals local ring dimension plus residue transcendence degree)

[F9]

A smooth finite-type $k$ -scheme has regular local rings. A localization $k [t, s] / (s (t^{2} - 1) - 1)$ is standard smooth over $k$ when the derivative with respect to $s$ is a unit. (Smoothness over a field by geometric regularity, Smooth morphisms via local standard smooth presentations, Standard smooth presentations and locally standard smooth maps, embedding dimension and regular local ring)

[F10]

Every affine scheme is separated over its base. (Affine schemes and affine morphisms are separated)

Refutation

Construction (nodal cubic). Let $k$ be a field of characteristic $\neq 2$ . Put $A : = k [x, y] / (y^{2} - x^{2} (x + 1)), X : = Spec A,$ and let $o \in X$ be the point $m = (x, y)$ . Let $\tilde{C} : = Spec k [t] = A^{1}$ and let $ν : \tilde{C} \to X$ be the $k$ -morphism with $ν^{♯} (x) = t^{2} - 1$ , $ν^{♯} (y) = t (t^{2} - 1)$ . Let $P^{1} = P_{k}^{1}$ with chart $U_{0} = Spec k [\hat{t}]$ .

Verification.

1.1F7

Geometric integrality over every field extension. Let $K / k$ be any field extension. In $K (x)$ , the order valuation $v_{x + 1}$ gives $v_{x + 1} (x^{2} (x + 1)) = 1$ because $x$ is a unit at the prime $(x + 1)$ of $K [x]$ ; a square has even valuation, so $x^{2} (x + 1)$ is not a square in $K (x)$ . Since $char K \neq 2$ , the quadratic $y^{2} - x^{2} (x + 1)$ has no root and is irreducible in $K (x) [y]$ . It is monic, hence primitive, over the UFD $K [x]$ , so Gauss's lemma makes it irreducible in $K [x, y]$ [F7]. The ring $K [x, y]$ is a UFD, so this irreducible polynomial is prime. Thus $K [x, y] / (y^{2} - x^{2} (x + 1))$ is a nonzero domain and its spectrum is integral Integral schemes. This holds for every extension $K / k$ , in particular for $\overset{ˉ}{k} / k$ , so $X$ is geometrically integral.

1.2F8F9

The point $o$ is closed since $A / m = k$ . Moreover $A / (x) ≅ k [y] / (y^{2})$ has the unique prime $(y)$ , so $D (x) = X ∖ {o}$ . The local ring at $o$ has dimension one: every open neighbourhood of the closed point $o$ contains the generic point of the integral curve $X$ and hence has chain dimension one, and [F8] identifies that local dimension with $\dim O_{X, o}$ . The maximal ideal modulo its square has basis given by the classes of $x, y$ , since $y^{2} - x^{2} (x + 1) \in (x, y)^{2}$ . Therefore $edim O_{X, o} = 2 \neq 1 = \dim O_{X, o}$ , so $O_{X, o}$ is not regular and $X$ is not smooth at $o$ embedding dimension and regular local ring and [F9].

1.3F8F9

The principal open $U : = D (x)$ is smooth. Set $t = y / x \in A_{x}$ ; the relation gives $x = t^{2} - 1$ and $y = t (t^{2} - 1)$ , so $A_{x} ≅ k [t, (t^{2} - 1)^{- 1}] ≅ k [t, s] / (s (t^{2} - 1) - 1)$ . The derivative with respect to $s$ is the unit $t^{2} - 1$ , so this is standard smooth over $k$ by [F9]. Since $D (x)$ is the complement of the singular point $o$ , the smooth locus of $X$ is exactly $U$ .

1.4F2F4F5

The rational map. The regular function $y / x$ defines a $k$ -morphism $φ_{U} : U \to P^{1}$ into $U_{0}$ by [F4]. Let $φ : X ⇢ P^{1}$ be represented by $(U, φ_{U})$ . The target is proper and separated over $k$ by [F5], so it satisfies the target hypotheses of [F2].

2.1F1F8step 1.1

Curve and dimension. For $K = k$ , the injection $k [x] ↪ A$ follows because a nonzero polynomial in $x$ cannot be divisible by the degree-two polynomial in $y$ . Hence $x$ is transcendental over $k$ and $y$ is algebraic over $k (x)$ , so ${trdeg}_{k} Frac (A) = 1$ . By [F8], $\dim A = 1$ and $Spec A$ has chain dimension one. The ring $A$ is finite type over $k$ , its affine structure morphism is separated [F10], and it is nonempty and geometrically integral by step 1.1. Thus $X$ is a curve over $k$ [F1, F8].

2.2F4step 1.4

The branch parameterization is a morphism. The ring map $A \to k [t]$ , $x \mapsto t^{2} - 1$ , $y \mapsto t (t^{2} - 1)$ is well defined because $t^{2} (t^{2} - 1)^{2} - (t^{2} - 1)^{3} - (t^{2} - 1)^{2} = 0$ , and it induces $ν$ . Both $t = 1$ and $t = - 1$ map to $o$ ; on $t^{2} - 1 \neq 0$ , $φ_{U} \circ ν$ is the chart-coordinate map $τ : A^{1} \to P^{1}$ given by $\hat{t} = t$ .

3.1F2F3step 2.2

Suppose a morphism $ψ : X \to P^{1}$ extends $φ$ . By rational-map equivalence, $ψ ∣_{U}$ and $φ_{U}$ agree on a nonempty open of the integral scheme $U$ , hence on a dense open. Since $U$ is reduced and $P^{1}$ is separated, [F3] gives $ψ ∣_{U} = φ_{U}$ . Thus $ψ \circ ν$ and $τ$ agree on $A^{1} ∖ {1, - 1}$ , a dense open of the reduced scheme $A^{1}$ ; [F3] gives $ψ \circ ν = τ$ everywhere.

4.1step 3.1

Evaluating at $1$ and $- 1$ gives $ψ (o) = τ (1)$ and $ψ (o) = τ (- 1)$ . The chart-coordinate points $\hat{t} = 1$ and $\hat{t} = - 1$ are distinct because $char k \neq 2$ , a contradiction. Therefore $φ$ has no extension.

5.1A1A2F1F3F6step 2.1step 1.2step 4.1∎

Any two extensions of $φ$ agree on the dense open $U$ of the reduced scheme $X$ ; since $P^{1}$ is separated, [F3] makes them equal. This disproves existence while preserving conditional uniqueness. It is an integral singular curve with a proper target, so the smoothness hypothesis in [F6] cannot be dropped. Choice availability is accounted for here: AC [A1] supplies DC [A2], the premise used by [F6] for the smooth-source comparison; this availability note is not part of the explicit two-branch contradiction.

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Divisor degree with residue degrees over a nonclosed field

Example

Assume the Axiom of Choice (The Axiom of Choice), inherited from the DVR local-ring context in Divisors on a smooth proper curve. Let $k = R$ and let $C = P_{R}^{1}$ with coordinate $t$ on the standard chart (Relative projective space from standard charts, Two-affine projective line and its twists). The closed point $x = V (t^{2} + 1)$ has residue field $κ (x) = R [t] / (t^{2} + 1) ≅ C$ , of degree two over $R$ , so the divisor $D = [x]$ satisfies $\deg_{R} (D) = 1 \cdot 2 = 2$ even though its support is a single point: the degree of a divisor weights each closed point by its residue degree (Degree divisor proper curve). The rational function $f = t^{2} + 1 \in R (C)^{\times}$ has divisor $div (f) = [x] - 2 [\infty],$ so $[x]$ is linearly equivalent to $2 [\infty]$ and the principal divisor has degree $2 - 2 \cdot 1 = 0$ , as it must. After base change to $C$ the point $x$ splits as the two $C$ -points $t = i$ and $t = - i$ , each of residue degree one over $C$ , with total degree $2 = \deg_{R} [x]$ .

Facts & Assumptions

Given: $k = R$ , the curve $C = P_{R}^{1}$ with coordinate $t$ on the standard affine chart $U_{0} = Spec R [t]$ , the closed point $x = V (t^{2} + 1) \subseteq U_{0}$ , the rational function $f = t^{2} + 1$ , and the divisor $D = [x]$ .

[F1]

A curve over a field $k$ is geometrically integral, separated and finite type of chain dimension one. Under AC, $P_{k}^{1}$ is a smooth proper geometrically integral curve for every field $k$ , hence for $k = R$ and $k = C$ . (Curves over a field, Projective-line curve and divisor basics)

[F2]

For a proper curve $C$ over $k$ , a divisor is a finite $Z$ -linear combination $D = \sum_{x} n_{x} [x]$ of closed points, the residue field $κ (x)$ of a closed point is a finite extension of $k$ , and $\deg_{k} D = \sum_{x} n_{x} [κ (x) : k]$ ; the degree is additive. (Degree divisor proper curve, Divisors on a smooth proper curve)

[F3]

The projective line $P_{k}^{1}$ has the two standard charts $U_{0} = Spec k [t]$ and $U_{\infty} = Spec k [u]$ glued along $t u = 1$ , with $\infty$ the origin $u = 0$ of the second chart; on a smooth curve the closed points are the maximal ideals of the chart rings. (Two-affine projective line and its twists, Curves over a field)

[F4]

Assume AC, inherited from the projective-line charts and curve basics in [F1] and [F3], as well as the smooth-curve DVR context. The closed-point local rings are DVRs, supplying the local orders in a principal divisor. The degree homomorphism itself is the choice-free finite sum in [F2]. (The Axiom of Choice, Divisors on a smooth proper curve, Degree divisor proper curve)

Proof

technique · direct; compute the residue field, the local orders of $f=t^2+1$ at its zero and at infinity, and the base change to $\mathbb C$

1.1F1F3

Residue field of $x$ . On the chart $U_{0} = Spec R [t]$ the point $x = V (t^{2} + 1)$ corresponds to the maximal ideal $(t^{2} + 1)$ , which is maximal because $t^{2} + 1$ is irreducible over $R$ (it has no real root and degree two); hence $κ (x) = R [t] / (t^{2} + 1) ≅ C$ [F3], a finite extension of $R$ of degree $2$ .

1.2F3

Order of vanishing at $x$ . In the local ring $O_{C, x} = R [t]_{(t^{2} + 1)}$ the element $t^{2} + 1$ generates the maximal ideal, hence is a uniformizer and ${ord}_{x} (f) = 1$ : the divisor of $f$ has the term $+ [x]$ .

1.3F3

Order of the pole at infinity. In the chart $U_{\infty} = Spec R [u]$ with $u = 1 / t$ one has $f = t^{2} + 1 = u^{- 2} (1 + u^{2})$ with $1 + u^{2}$ a unit of the local ring at $u = 0$ because it evaluates to $1$ there; hence ${ord}_{\infty} (f) = - 2$ and the divisor of $f$ has the term $- 2 [\infty]$ .

2.1F2step 1.1

Degree of $[x]$ . By the degree formula of [F2], $\deg_{R} ([x]) = 1 \cdot [κ (x) : R] = 1 \cdot 2 = 2$ , while the support of $[x]$ is the single point $x$ ; this is the sense in which the degree counts with residue-field degrees rather than with a point count.

3.1F2F3step 1.2step 1.3step 2.1

Principal divisor. At every other closed point of $U_{0}$ , $t^{2} + 1$ is a unit, since its only irreducible factor is $t^{2} + 1$ ; the complement of $U_{0}$ is the single point $\infty$ . Thus steps 1.2 and 1.3 account for every nonzero order: $div (f) = [x] - 2 [\infty]$ , and its degree is $\deg_{R} ([x]) - 2 \deg_{R} ([\infty]) = 2 - 2 \cdot 1 = 0$ by [F2]; the point $\infty$ has residue field $R$ and degree one. Thus $[x]$ is linearly equivalent to $2 [\infty]$ , a divisor of the same degree $2$ .

3.2F2step 2.1

Base change to $C$ . On the base-changed affine chart, the fibre of $x$ has coordinate algebra $C [t] / (t^{2} + 1) = C [t] / ((t - i) (t + i))$ . Evaluation at $i$ and $- i$ identifies this algebra with $C \times C$ : every class has a unique representative $a + b t$ , and its evaluations $a + b i, a - b i$ determine $a, b$ uniquely. Thus the fibre consists of the two distinct reduced points $t = i$ and $t = - i$ , each with residue field $C$ and degree one. At each point $t^{2} + 1$ has order one, since its other linear factor is a unit. Hence the base-changed divisor is $[i] + [- i]$ , of degree $2 = \deg_{R} ([x])$ .

4.1F2F4step 2.1step 3.2∎

Conclusion. On $P_{R}^{1}$ the divisor $D = [x]$ has degree $2$ although it is supported at one point, its class is the class of $2 [\infty]$ by the principal divisor $[x] - 2 [\infty]$ , and after base change to $C$ it becomes the sum of the two degree-one points $i, - i$ with the same total degree. Degree is therefore computed with residue-field degrees, as in [F2]; Choice in [F4] is inherited from the projective-line and DVR suppliers, whereas additivity of the degree is choice-free and no further selection is used here.

ExampleConstruction: AI-generatedVerification: AI-adaptedaudited 2026-10-02Open item page →

A linear system with and without a base point

Example

Assume the Axiom of Choice as inherited from the projective-space constructions (The Axiom of Choice). On $P_{k}^{1}$ with coordinate $t$ and $D = 2 [\infty]$ (Two-affine projective line and its twists, Divisors on a smooth proper curve), put $V = span (1, t^{2}) \subseteq L (D), W = span (t, t^{2}) \subseteq L (D) .$ Both are two-dimensional subspaces of $L (D)$ (The space L(D)), so base-point-freeness is a property of the chosen subsystem and not of its degree. The subspace $V$ is base-point-free and defines the degree-two morphism $φ_{V} : P_{k}^{1} \to P_{k}^{1}$ , $[1 : t] \mapsto [1 : t^{2}]$ ; the subspace $W$ has $t = 0$ as its unique base point, and its associated rational map is $[t : t^{2}] = [1 : t]$ on $P_{k}^{1} ∖ {0}$ , which extends to the identity morphism of $P_{k}^{1}$ . The pair $(t, t^{2})$ does not generate $O (D)$ at its base point, so the base-point-free construction for this line bundle does not apply to that pair. In the set identification $∣ D ∣ ≅ P (L (D)) ≅ P_{k}^{2} (k)$ of Divisors and complete linear systems on the projective line, use coordinates $(a, b, c)$ for $a + b t + c t^{2}$ . The associated projective parameter scheme is $P (L (D)) ≅ P_{k}^{2}$ ; the intersections below are scheme intersections in this parameter scheme. If $char k \neq 2$ , the discriminant conic $b^{2} = 4 a c$ is smooth, $P (W) : a = 0$ is tangent to it at $2 [0]$ , and $P (V) : b = 0$ is the secant through $2 [0]$ and $2 [\infty]$ . If $char k = 2$ , the discriminant scheme is the double line $b^{2} = 0$ ; its reduced support $b = 0$ is the geometric doubled-divisor locus, and the coordinate-square map $[T : S] \mapsto [T^{2} : 0 : S^{2}]$ is onto that support as a morphism. Then $P (V)$ is the support line, while $P (W)$ meets it at $2 [0]$ and meets the double discriminant in a length-two point. Over an imperfect field, not every $k$ -point of the support need come from a $k$ -rational doubled divisor.

Facts & Assumptions

Given: A field $k$ , the projective line $P_{k}^{1}$ with coordinate $t$ on $U_{0} = Spec k [t]$ , the point at infinity $\infty$ the pole of $t$ , the divisor $D = 2 [\infty]$ , and the two subspaces $V = span (1, t^{2})$ , $W = span (t, t^{2})$ of $L (D)$ .

[F1]

On $P_{k}^{1}$ one has $div (t) = [0] - [\infty]$ and, for a nonzero polynomial $p$ of degree $m$ , $div (p) = Z (p) - m [\infty]$ , where $Z (p)$ is the effective divisor of the affine zeros of $p$ ; $\deg_{k} (2 [\infty]) = 2$ , and the constant function $1$ has $div (1) = 0$ . (Divisors and complete linear systems on the projective line, Divisors on a smooth proper curve, Degree divisor proper curve)

[F2]

$L (D) = {f \in k (P^{1})^{\times} : div (f) + D \geq 0} \cup {0}$ is the $k$ -subspace of rational functions with poles bounded by $D$ . (The space L(D))

[F3]

A nonzero $f \in L (D)$ vanishes at a closed point $x$ when ${ord}_{x} (f) + n_{x} \geq 1$ , where $n_{x}$ is the coefficient of $D$ at $x$ ; $x$ is a base point of a subspace $V \subseteq L (D)$ when every nonzero $f \in V$ vanishes at $x$ ; $V$ is base-point-free when it has no base point, equivalently when the evaluation morphism of a basis of $V$ is surjective. (Base points and base-point-free linear systems)

[F4]

A base-point-free subspace $V \subseteq L (D)$ of dimension $r + 1 \geq 1$ determines a $k$ -morphism $φ_{V} : P_{k}^{1} \to P_{k}^{r}$ with $φ_{V}^{*} O (1) ≅ O (D)$ under which the coordinate sections pull back to a basis of $V$ ; for generating sections $s_{0}, \dots, s_{r}$ the chart formula $x_{j}^{(i)} \circ φ = s_{j} / s_{i}$ holds on the locus where $s_{i}$ is invertible, and two morphisms to the separated reduced $k$ -scheme $P_{k}^{r}$ agreeing on a dense open are equal. (A base-point-free linear system defines a morphism to projective space, Generating line-bundle sections define a morphism to projective space, Agreement on a schematically dense open)

[F5]

For $d \geq 0$ the complete linear system $∣ d [\infty] ∣$ is in bijection with $P (L (d [\infty]))$ , the set of $k$ -lines in $L (d [\infty])$ , and consists of the effective divisors of degree $d$ ; for $D = 2 [\infty]$ the associated morphism of the complete system is the degree-two Veronese map $ν_{1, 2} : P_{k}^{1} \to P_{k}^{2}$ , $[x_{0} : x_{1}] \mapsto [x_{0}^{2} : x_{0} x_{1} : x_{1}^{2}]$ , and $∣ D ∣ ≅ P (L (D)) ≅ P_{k}^{2} (k)$ . The projective parameter scheme $P (L (D)) ≅ P_{k}^{2}$ has this set of $k$ -points, with coefficient coordinates $(a, b, c)$ for $a + b t + c t^{2}$ ; the inclusions of $V, W$ give projective linear subschemes $P (V), P (W)$ in it. (Divisors and complete linear systems on the projective line, Complete linear system)

[F6]

The divisor-doubling map in coefficient coordinates is $[T : S] \mapsto [T^{2} : - 2 T S : S^{2}]$ and lies in the discriminant scheme $b^{2} = 4 a c$ . If $char k \neq 2$ , its image is the smooth conic, and its $k$ -points are exactly the squares of linear forms up to nonzero scalar. If $char k = 2$ , the discriminant scheme is $b^{2} = 0$ , a double line with reduced support $b = 0$ ; the doubling map becomes $[T : S] \mapsto [T^{2} : 0 : S^{2}]$ and has that reduced line as its scheme-theoretic image. It is surjective onto the support geometrically, while its image on $k$ -points can be smaller over an imperfect field. (Divisors and complete linear systems on the projective line, Complete linear system)

[F7]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

technique · direct; test the two subspaces against the vanishing criterion, compute their chart maps, and read their pencils off the model of the projective line

1.1F1F2F5

The two subspaces. By [F1], $div (1) + 2 [\infty] = 2 [\infty] \geq 0$ , $div (t) + 2 [\infty] = [0] + [\infty] \geq 0$ , $div (t^{2}) + 2 [\infty] = 2 [0] \geq 0$ , so $1, t, t^{2} \in L (2 [\infty])$ and $V, W$ are subspaces of $L (D)$ ; the pairs $(1, t^{2})$ and $(t, t^{2})$ are linearly independent over $k$ , so $\dim_{k} V = \dim_{k} W = 2$ and $V, W$ are two-dimensional subsystems of the degree-two complete system $∣ D ∣$ .

1.2F1F3F4

$V$ is base-point-free and defines $t \mapsto t^{2}$ . Since $div (1) + 2 [\infty] = 2 [\infty]$ is supported at $\infty$ , the function $1$ does not vanish at any closed point $x \neq \infty$ , so no point other than possibly $\infty$ is a base point of $V$ ; and since $div (t^{2}) + 2 [\infty] = 2 [0]$ is supported at the origin, $t^{2}$ does not vanish at $\infty$ , so $\infty$ is not a base point either [F3]. Hence $V$ is base-point-free, and [F4] attaches to $V$ a morphism $φ_{V} : P_{k}^{1} \to P_{k}^{1}$ with $φ_{V}^{*} O (1) ≅ O (D)$ , whose coordinate sections pull back to $1, t^{2}$ ; on the chart where $1$ is invertible, which is $P_{k}^{1} ∖ {\infty}$ , the chart formula gives $φ_{V} ([1 : t]) = [1 : t^{2}]$ , the degree-two map $t \mapsto t^{2}$ .

1.3F1F3F4

$W$ has the base point $0$ , but its rational map extends. Every nonzero $f = α t + β t^{2} = t (α + β t) \in W$ has ${ord}_{0} (f) \geq 1$ while the coefficient of $D$ at $0$ is $n_{0} = 0$ , so every such $f$ vanishes at the origin [F3]; and no other point is a base point, since $t$ does not vanish at closed points $x \neq 0, \infty$ while $t^{2}$ does not vanish at $\infty$ [F1, F3]. Hence the base locus of $W$ is exactly ${0}$ , and its two sections do not generate $O (D)$ there, so [F4]'s base-point-free construction does not attach a morphism from this pair with pullback line bundle $O (D)$ . On the complement of $0$ , however, the pair defines $[t : t^{2}] = [1 : t]$ , the identity rational map, which extends to the identity morphism of $P_{k}^{1}$ . The rational map extension is unique because morphisms to the separated target $P_{k}^{1}$ agreeing on a dense open are equal [F4].

2.1F5F6step 1.1step 1.2step 1.3

The pencil picture in the parameter scheme $P (L (D)) ≅ P_{k}^{2}$ . By [F5], this scheme has coefficient coordinates $(a, b, c)$ for $a + b t + c t^{2}$ , and the subspaces correspond to $P (V) : b = 0$ and $P (W) : a = 0$ . The divisor-doubling map of [F6] is $[T : S] \mapsto [T^{2} : - 2 T S : S^{2}]$ ; its coefficient image satisfies $b^{2} = 4 a c$ , with $2 [0] = (0, 0, 1)$ and $2 [\infty] = (1, 0, 0)$ . If $char k \neq 2$ , this is a smooth conic. On $P (W)$ , $a = 0$ forces $b^{2} = 0$ , so the intersection is the length-two point $2 [0]$ and $P (W)$ is tangent there. On $P (V)$ , $b = 0$ forces $a c = 0$ , giving the two distinct points $2 [0]$ and $2 [\infty]$ ; thus $P (V)$ is a secant. In this characteristic, a pencil has a base point exactly when its line is tangent to the doubled-divisor conic, consistent with steps 1.2 and 1.3.

If $char k = 2$ , the discriminant scheme is $b^{2} = 0$ , the double of the reduced support line $b = 0$ . The doubling map becomes $[T : S] \mapsto [T^{2} : 0 : S^{2}]$ ; on either standard affine chart its coordinate map is $w \mapsto w^{2}$ , so it is finite and surjective onto that support as a morphism, although it need not be onto its $k$ -points when $k$ is imperfect. Thus $P (V)$ is the reduced support of the geometric doubled-divisor locus, while $P (W)$ meets that support at $2 [0]$ . Its intersection with the double discriminant has local ring $k [b] / (b^{2})$ at that point and therefore length two. The characteristic-not-two tangent/secant description is not asserted in characteristic two. [F6]

3.1F4F7step 1.2step 1.3step 2.1∎

Conclusion. On $P_{k}^{1}$ with $D = 2 [\infty]$ , $V = span (1, t^{2})$ is base-point-free and defines $[1 : t] \mapsto [1 : t^{2}]$ , while $W = span (t, t^{2})$ has the single base point $0$ and its rational map extends to the identity morphism (steps 1.2–1.3). Both subsystems have degree two, so base-point-freeness depends on the chosen subsystem and not its degree. Their pencil geometry is the tangent/secant picture of step 2.1 in characteristic not two; in characteristic two, $P (V)$ is the reduced support of the double discriminant and $P (W)$ meets the doubled scheme in a length-two point. The Axiom of Choice is inherited from the projective-space constructions of [F4] and [F7], and no further selection is used.

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A nontrivial degree-zero line bundle has no nonzero section

Counterexample

Assume the Axiom of Choice (The Axiom of Choice), inherited from the Cartier-Weil, genus, finite-map and function-field suppliers. Let $k$ be an algebraically closed field and let $E = V_{+} (y^{2} z - x^{3} - a x z^{2} - b z^{3}) \subseteq P_{k}^{2}$ be a smooth plane cubic, a curve of genus one (Arithmetic genus of a plane curve), with distinct closed points $P, Q \in E$ . Then the invertible sheaf $L = O_{E} (P - Q)$ (Invertible sheaf of cartier divisor) has degree zero, $H^{0} (E, L) = 0$ , and $L$ is nontrivial: a nonzero section would present $L$ as $O_{E} (D)$ with $D$ effective and $\deg D = 0$ , forcing $D = 0$ and $L ≅ O_{E}$ , while nontriviality holds because $O_{E} (P - Q) ≅ O_{E}$ would give a rational function of divisor $P - Q$ and hence a degree-one map $E \to P^{1}$ , impossible for a curve of genus one. So degree zero neither forces triviality nor produces sections.

Facts & Assumptions

Given: An algebraically closed field $k$ , the smooth plane cubic $E = V_{+} (F)$ with $F = y^{2} z - x^{3} - a x z^{2} - b z^{3}$ , and distinct closed points $P, Q \in E$ .

[F1]

Let $X = V_{+} (G) \subseteq P_{k}^{2}$ be cut out by a nonzero homogeneous form of degree $d \geq 1$ and assume $X$ is a curve (integral of dimension one); then $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = 1 - χ (O_{X}) = \frac{(d - 1) (d - 2)}{2}$ . In particular a smooth plane cubic is an integral proper curve with $p_{a} = 1$ , and a line has $p_{a} = 0$ . (Arithmetic genus of a plane curve)

[F2]

A curve over $k$ is geometrically integral, separated, finite type of chain dimension one; a smooth proper curve is in particular integral and reduced, and the arithmetic genus $p_{a} (X) = 1 - χ (O_{X})$ of an integral proper curve is an invariant of its isomorphism class, because the cohomology of the structure sheaf depends only on the scheme up to isomorphism. (Curves over a field, Genus and arithmetic genus of a curve)

[F3]

On a smooth curve divisors are finite $Z$ -combinations of closed points, $\deg_{k} (\sum_{x} n_{x} [x]) = \sum_{x} n_{x} [κ (x) : k]$ , and over an algebraically closed field every closed point has residue field $k$ and degree one. (Divisors on a smooth proper curve, Degree divisor proper curve)

[F4]

For a smooth proper geometrically integral curve $C$ : the group $Pic (C)$ of isomorphism classes of invertible sheaves is identified with the divisor class group by $[D] \mapsto [O_{C} (D)]$ , the degree $\deg_{k}$ descends to a homomorphism $Pic (C) \to Z$ taking $O_{C} (D)$ to $\deg_{k} D$ , and $O_{C} (D) ≅ O_{C} (D^{'})$ if and only if $D$ and $D^{'}$ are linearly equivalent; the sheaf $O_{C} (D)$ is the invertible sheaf attached to the Cartier divisor $D$ . (Cartier and Weil divisors agree on a smooth curve, The degree of a divisor descends to the Picard group of a normal proper curve, Invertible sheaf of cartier divisor)

[F5]

A nonzero rational section $s$ of an invertible sheaf $M$ on a smooth curve $C$ determines a divisor $div (s)$ with $M ≅ O_{C} (div (s))$ carrying the canonical section $1$ to $s$ ; consequently nonzero global sections of $M$ correspond to effective divisors $D$ with $O_{C} (D) ≅ M$ . (Rational sections of line bundles are Cartier divisors, Effective divisors linearly equivalent to D are sections modulo scalars)

[F6]

If $D$ is an effective divisor on a proper geometrically integral curve over $k$ , then $\deg_{k} D \geq 0$ , and $\deg_{k} D = 0$ if and only if $D = 0$ . (Effective divisors have nonnegative degree)

[F7]

A nonconstant rational function $f \in k (C)^{\times}$ on a smooth proper geometrically integral curve defines a finite morphism $φ_{f} : C \to P_{k}^{1}$ whose degree is $[k (C) : k (f)]$ and whose fibre over infinity is the pole divisor $(f)_{\infty}$ , of the same degree. (A nonconstant rational function defines a finite map to the projective line, Degree of a nonconstant morphism of curves)

[F8]

Under Choice the assignment $f \mapsto f^{*}$ is a bijection from dominant morphisms between smooth proper geometrically integral curves over $k$ onto injective $k$ -algebra homomorphisms of function fields, and every birational rational map between smooth proper geometrically integral curves is represented by a $k$ -isomorphism. (Smooth proper curves, dominant morphisms and function fields, Birational smooth proper curves are isomorphic)

[F9]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

[F10]

Polynomial rings in finitely many variables over a field are UFDs; irreducibles are prime. A nonzero positive-degree plane hypersurface has pure dimension one. Closed immersions and projective-space structure maps are proper, and proper morphisms compose. (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Nontrivial projective hypersurface sections, Finite-dimensional projective space is proper over every base, Closed immersions are proper, Properness survives composition)

Proof

technique · direct; show the curve has $p_a=1$ while a trivial class would force a degree-one map to $\mathbb P^1$

1.1F1F2F10givenalgebra

Integrality and genus. On $z = 1$ the equation is $y^{2} - (x^{3} + a x + b)$ . The cubic polynomial in $x$ is not a square in $k (x)$ : its order at infinity is $- 3$ , while a square has even order. Hence the monic quadratic in $y$ is irreducible over $k (x)$ and, by clearing denominators in the UFD $k [x]$ , over $k [x, y]$ . Its homogenization $F$ is irreducible too: a nonconstant homogeneous factor becoming constant at $z = 1$ would be a scalar power of $z$ , but $z$ does not divide $F$ . Thus [F10] makes the homogeneous quotient a domain; its nonempty projective charts are domains with common generic point, so $E$ is integral. It is of dimension one and proper by [F10]. Since $k$ is algebraically closed it is geometrically integral, and it is smooth by hypothesis, so [F1] applies with $d = 3$ to give $H^{0} (E, O_{E}) = k$ and $p_{a} (E) = 1$ .

1.2F3F4

The class has degree zero. The closed points $P, Q$ are $k$ -rational, so $[κ (P) : k] = [κ (Q) : k] = 1$ by [F3]; hence $\deg_{k} (P - Q) = \deg_{k} (P) - \deg_{k} (Q) = 1 - 1 = 0$ , and by [F4] the invertible sheaf $L = O_{E} (P - Q)$ has $\deg (L) = \deg_{k} (P - Q) = 0$ .

2.1F1F2F4F7F8step 1.1

The class is nontrivial. Suppose $O_{E} (P - Q) ≅ O_{E}$ ; by [F4] this means that $P - Q = div (f)$ for a rational function $f \in k (E)^{\times}$ , so the pole divisor $(f)_{\infty}$ of $f$ is the single point $Q$ with multiplicity one and the divisor of $f$ is nonzero; in particular $f$ is nonconstant, and by [F7] it defines a morphism $φ_{f} : E \to P_{k}^{1}$ of degree $[k (E) : k (f)] = \deg_{k} (f)_{\infty} = 1$ . Consequently $k (E) = k (f)$ is a degree-one extension of $k (f)$ , so $φ_{f}$ is birational as a morphism of integral curves and [F8] represents it by a $k$ -isomorphism $E ≅ P_{k}^{1}$ . The arithmetic genus is an isomorphism invariant [F2], so $p_{a} (E) = p_{a} (P_{k}^{1}) = 0$ by [F1] applied to a line, contradicting $p_{a} (E) = 1$ from step 1.1. Hence $L$ is nontrivial.

3.1F4F5F6step 1.2step 2.1

Every nonzero section forces triviality. Suppose $s \in H^{0} (E, L)$ is nonzero. By [F5] there is an effective divisor $D = div (s)$ on $E$ with $O_{E} (D) ≅ L$ ; by [F4] the degree of $L$ is $\deg_{k} D$ , which is $0$ by step 1.2. So $D$ is an effective divisor of degree zero, and [F6] forces $D = 0$ ; then $L ≅ O_{E} (D) ≅ O_{E}$ , contradicting the nontriviality of $L$ from step 2.1. Therefore $H^{0} (E, L) = 0$ .

4.1F9step 1.2step 2.1step 3.1∎

Conclusion. For distinct closed points $P, Q$ on the smooth plane cubic $E$ of genus one, the invertible sheaf $L = O_{E} (P - Q)$ has degree $\deg_{k} (P - Q) = 0$ by steps 1.2, is nontrivial by step 2.1, and has no nonzero global section by step 3.1: degree zero neither forces a line bundle to be trivial nor guarantees that it has a section. The Axiom of Choice is inherited from the Cartier-Weil, genus, finite-map and function-field suppliers [F9] and no further choice is used.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-10-02Open item page →

Ramification indices of the power map on the projective line

Example

Assume the Axiom of Choice (The Axiom of Choice), inherited from the finite morphism to the projective line and from the divisor theory of the projective line. Let $k$ be a field, let $n \geq 1$ be an integer with $char k ∤ n$ , and let $P_{k}^{1}$ have homogeneous coordinates $[s : t]$ , origin $0 = [1 : 0]$ , point at infinity $\infty = [0 : 1]$ , and affine coordinate $x = t / s$ on the chart $U_{0} = {s \neq 0}$ with $y = s / t = x^{- 1}$ on the chart $U_{\infty} = {t \neq 0}$ . Let $φ ⁣ : P_{k}^{1} \to P_{k}^{1}$ be the morphism given in these coordinates by $φ ([s : t]) = [s^{n} : t^{n}]$ , so that on the charts it is $x \mapsto x^{n}$ and $y \mapsto y^{n}$ . Then:

$φ$ is a finite surjective morphism of smooth proper geometrically integral curves of degree $\deg (φ) = n$ ;
at every closed point $p \notin {0, \infty}$ the morphism is unramified: $e_{p} = 1$ and the residue extension $κ (p) / κ (φ (p))$ is separable; the index-ramification locus of $φ$ is ${0, \infty}$ when $n \geq 2$ and is empty when $n = 1$ , and the differential-ramification locus is likewise ${0, \infty}$ when $n \geq 2$ and empty when $n = 1$ ;
at $0$ and at $\infty$ the fibre of $φ$ is a single point and the ramification index is $n$ : the pullback of a uniformizer of the target at the image point has order exactly $n$ , and the fibre degree sum reads $n = n$ over each of the two points;
the tame case is the case at hand, because $char k ∤ n$ ; the ramification divisor $R_{φ} = \sum_{p} l_{p} [p]$ , where $l_{p}$ is the length over $O_{P_{k}^{1}, p}$ of the relative differentials $Ω_{P_{k}^{1} / P_{k}^{1}}$ at $p$ , equals $R_{φ} = (n - 1) ([0] + [\infty])$ : it has support ${0, \infty}$ with length $n - 1$ at each point when $n \geq 2$ , and it is the zero divisor when $n = 1$ .

Supplier interfaces. The current draft Projective-line curve and divisor basics supplies the projective-line and divisor facts in [F1]. The current draft A nonconstant rational function defines a finite map to the projective line supplies the map and its zero/pole fibre identifications; this example computes the degree independently from Fibre degree sum with ramification and residue degrees, so it does not use the separate Fibre degree of the finite locally free map to the projective line calculation.

Facts & Assumptions

Given: A field $k$ , an integer $n \geq 1$ with $char k ∤ n$ , the projective line $P_{k}^{1}$ with charts $U_{0} = Spec k [x]$ and $U_{\infty} = Spec k [y]$ , $y = x^{- 1}$ , points $0 = V (x)$ and $\infty = V (y)$ , and the morphism $φ ⁣ : P_{k}^{1} \to P_{k}^{1}$ with $φ^{♯} (x) = x^{n}$ on the target coordinate $x$ ; the Axiom of Choice is assumed.

[F1]

The projective line $P_{k}^{1}$ is a smooth proper geometrically integral curve over $k$ with standard charts $U_{0} = Spec k [x]$ , $U_{\infty} = Spec k [y]$ glued along $x y = 1$ ; its closed points in $U_{0}$ are the points $V (g)$ for monic irreducible $g \in k [x]$ , with $[κ (V (g)) : k] = \deg g$ , and the divisor of the rational function $g$ is $div (g) = [V (g)] - (\deg g) [\infty]$ ; in particular $div (x) = [0] - [\infty]$ , the points $0$ and $\infty$ are $k$ -rational, and the local ring $O_{P_{k}^{1}, p}$ at a closed point $p \in U_{0}$ is the localization $k [x]_{(h)}$ at the maximal ideal $(h)$ defining $p$ . (Projective-line curve and divisor basics, Two-affine projective line and its twists, Relative projective space from standard charts, Curves over a field)

[F2]

A nonconstant rational function $f \in k (P_{k}^{1})^{\times}$ determines a finite locally free morphism $φ_{f} ⁣ : P_{k}^{1} \to P_{k}^{1}$ of degree $[k (P_{k}^{1}) : k (f)]$ with $φ_{f}^{♯} (x) = f$ for the target coordinate $x$ , whose fibre over $0$ is the zero divisor $(f)_{0} = \sum_{{ord}_{p} (f) > 0} {ord}_{p} (f) [p]$ and whose fibre over $\infty$ is the pole divisor $(f)_{\infty} = \sum_{{ord}_{p} (f) < 0} (- {ord}_{p} (f)) [p]$ ; in particular $φ_{f}$ is nonconstant and, being a morphism of proper curves, it is surjective. (A nonconstant rational function defines a finite map to the projective line, Degree of a nonconstant morphism of curves, Nonconstant morphisms of proper curves are finite and surjective)

[F3]

At a closed point $p$ of a smooth curve with image $q = φ (p)$ the local rings are discrete valuation rings, a uniformizer is a generator of the maximal ideal, every nonzero element is a unit times a power of a uniformizer, the order ${ord}_{p}$ is additive and vanishes on units, the ramification index is $e_{p} = {ord}_{p} (φ^{♯} (t_{q}))$ for a uniformizer $t_{q}$ of $O_{D, q}$ , and $e_{p} = 1$ exactly for the unramified points of the index convention. (Ramification index of a morphism of curves, Local rings at closed points of smooth curves are discrete valuation rings, Every nonzero fraction is a unit times a power of a uniformiser, Order codimension one rational function)

[F4]

For a nonconstant morphism $φ ⁣ : C \to D$ of smooth proper geometrically integral curves of degree $n$ and every closed point $q$ of $D$ the fibre is finite and $\sum_{p \in φ^{- 1} (q)} e_{p} [κ (p) : κ (q)] = n$ . (Fibre degree sum with ramification and residue degrees, Curves over a field)

[F5]

For a finite surjective morphism $φ ⁣ : C \to D$ of smooth proper geometrically integral curves with separable function-field extension the sheaf $Ω_{C / D}$ of relative differentials is coherent and torsion with finite support, and with $l_{p} = {length}_{O_{C, p}} (Ω_{C / D, p})$ one has $l_{p} = 0$ if and only if $e_{p} = 1$ and $κ (p) / κ (φ (p))$ is separable; if the residue extension is separable and $e_{p}$ is invertible in $κ (φ (p))$ , then $l_{p} = e_{p} - 1$ ; and the differential-ramification locus is the support of $Ω_{C / D}$ , which equals the set of points with $e_{p} > 1$ or inseparable residue extension. (Local support and index bound for the different of a curve map, Sheaf of relative Kähler differentials, Ramification points, branch points and unramifiedness, Composition series and length of a module)

[F6]

On an affine chart, if a morphism of affine schemes corresponds to the ring map $A \to B$ and $B = A [x_{1}, \dots, x_{r}] / I$ , then $Ω_{B / A}$ is the cokernel of the Jacobian map $B^{c} \to B^{r}$ of a set of generators of $I$ ; in particular for $B = A [x] / (g)$ one has $Ω_{B / A} ≅ B / (g^{'} (x))$ . Localizing at a multiplicative set computes the corresponding localization of the module, and for an affine open $U = Spec B$ of the source mapping into an affine open of the target the module of sections of $Ω_{C / D}$ over $U$ is $Ω_{B / A}$ . (Differentials of a polynomial quotient and the Jacobian cokernel, Kähler differentials commute with localization, Sheaf of relative Kähler differentials)

[F7]

For a discrete valuation ring $V$ with uniformizer $π$ and $n \geq 0$ one has $ℓ_{V} (V / π^{n}) = n$ , so a module with a filtration by powers of the uniformizer has length equal to the number of successive quotients. (Length and valuation in a DVR, Composition series and length of a module)

[F8]

A divisor on a curve is a finite formal $Z$ -linear combination of closed points and is effective when all coefficients are nonnegative; the divisors $[p]$ of closed points generate it. (Divisors on a smooth proper curve)

[F9]

The Axiom of Choice is assumed, here inherited from the construction of $φ$ as a finite morphism to the projective line and from the divisor theory of $P_{k}^{1}$ ; no further selection is made. (The Axiom of Choice)

Proof

technique · direct; exhibit $\varphi$ as the finite map attached to the rational function $x^n$, read off the ramification indices at $0$ and $\infty$ from the zero and pole divisors, determine the degree from the fibre-degree sum, and compute the relative differentials on the two affine charts

1.1F1F2

The morphism. The coordinate $x$ satisfies ${ord}_{\infty} (x) = - 1$ by [F1], so $x$ is nonconstant and hence $x^{n}$ is nonconstant as well. By [F2] applied to $f = x^{n}$ there is a finite locally free morphism $φ ⁣ : P_{k}^{1} \to P_{k}^{1}$ of degree $[k (P_{k}^{1}) : k (x^{n})]$ with $φ^{♯} (x) = x^{n}$ ; it is nonconstant, hence surjective. On the affine charts the comorphism is $k [x] \to k [x]$ , $x \mapsto x^{n}$ on $U_{0}$ , and $k [y] \to k [y]$ , $y = x^{- 1} \mapsto (x^{n})^{- 1} = y^{n}$ on $U_{\infty}$ . In the homogeneous coordinates of [F1] the target coordinate of the image of $[s : t]$ with $s \neq 0$ is $x^{n} = (t / s)^{n}$ , so the image is $[1 : t^{n} / s^{n}] = [s^{n} : t^{n}]$ ; thus $φ$ is the morphism of the statement.

1.2F1F2F3

Zeros and poles. The order function of [F3] is additive, so ${ord}_{p} (x^{n}) = n {ord}_{p} (x)$ for every closed point $p$ ; by [F1] the only points with ${ord}_{p} (x) \neq 0$ are $0$ , where ${ord}_{0} (x) = 1$ , and $\infty$ , where ${ord}_{\infty} (x) = - 1$ . Hence $div (x^{n}) = n [0] - n [\infty]$ , the zero divisor of $x^{n}$ is $(x^{n})_{0} = n [0]$ , and its pole divisor is $(x^{n})_{\infty} = n [\infty]$ . By [F2] the fibre of $φ$ over $0$ is carried by $n [0]$ and the fibre over $\infty$ by $n [\infty]$ ; in particular $φ^{- 1} (0) = {0}$ and $φ^{- 1} (\infty) = {\infty}$ as sets, with $κ (0) = κ (\infty) = k$ by [F1].

1.3F1F3

Ramification at $0$ and at $\infty$ . By [F1] the element $x$ is a uniformizer of $O_{P_{k}^{1}, 0}$ , and the pullback of the target uniformizer $x$ at $0$ is $φ^{♯} (x) = x^{n}$ , so $e_{0} = {ord}_{0} (x^{n}) = n$ by [F3]. At infinity $y = x^{- 1}$ is a uniformizer of $O_{P_{k}^{1}, \infty}$ by [F1], and the pullback of the target uniformizer $y$ at $\infty$ is $φ^{♯} (y) = y^{n}$ , so $e_{\infty} = {ord}_{\infty} (y^{n}) = n$ .

2.1F2F4step 1.2step 1.3

The degree is $n$ . The function field extension $k (P_{k}^{1}) / k (x^{n})$ is separable because $x$ satisfies the polynomial $T^{n} - x^{n} \in k (x^{n}) [T]$ whose derivative $n T^{n - 1}$ has no common root with it in characteristic not dividing $n$ , so [F4] applies to the nonconstant morphism $φ$ of degree $[k (P_{k}^{1}) : k (x^{n})]$ . Evaluating the fibre-degree sum of [F4] at $q = 0$ and using that the fibre is the single point $0$ with $κ (0) = k$ by step 1.2 and $e_{0} = n$ by step 1.3 gives $\deg (φ) = e_{0} [κ (0) : k] = n$ . The same computation at $\infty$ gives $\deg (φ) = e_{\infty} [κ (\infty) : k] = n$ , and the two readings agree.

2.2F1F5F6F7step 1.1

Relative differentials on the two charts. On $U_{0}$ the map of affine charts is the ring map $A = k [x] \to B = k [x]$ , $x \mapsto x^{n}$ , so $B = A [T] / (T^{n} - x)$ with $T \mapsto x$ and $g (T) = T^{n} - x$ ; its derivative is $g^{'} (T) = n T^{n - 1}$ , and the class $n \in k$ is a unit because $char k ∤ n$ , so [F6] gives $Ω_{B / A} ≅ B / (T^{n - 1}) = k [x] / (x^{n - 1})$ . Localizing at the maximal ideal $(h)$ of a closed point $p = V (h) \in U_{0}$ as in [F1] and using the localization clause of [F6], the stalk is $Ω_{P_{k}^{1} / P_{k}^{1}, p} ≅ k [x]_{(h)} / (x^{n - 1})$ : this is zero when $h \neq x$ , because then $x$ is a unit of the localization, and for $p = 0$ it is the module $k [x]_{(x)} / (x^{n - 1})$ , whose filtration by the powers of the uniformizer $x$ has $n - 1$ successive quotients isomorphic to $κ (0) = k$ , so its length is $n - 1$ by [F7]. The same computation in the coordinate $y$ on $U_{\infty}$ gives $Ω_{P_{k}^{1} / P_{k}^{1}, p} = 0$ for $p \in U_{\infty}$ with $p \neq \infty$ and length $n - 1$ at $\infty$ . Hence the relative differentials are supported exactly on ${0, \infty}$ , with $l_{0} = l_{\infty} = n - 1$ , and this support is empty exactly when $n = 1$ .

3.1F5step 1.3step 2.2

Unramifiedness away from $0$ and $\infty$ . Let $p \notin {0, \infty}$ be a closed point. By step 2.2 the stalk $Ω_{P_{k}^{1} / P_{k}^{1}, p}$ vanishes, so $l_{p} = 0$ , and the criterion of [F5] gives $e_{p} = 1$ together with separability of $κ (p) / κ (φ (p))$ ; in the terminology of [F5] the point $p$ is unramified and lies in neither the index-ramification locus nor the differential-ramification locus. Since $e_{0} = e_{\infty} = n$ by step 1.3, the index-ramification locus equals ${0, \infty}$ when $n \geq 2$ and is empty when $n = 1$ , and by step 2.2 the same holds for the differential-ramification locus.

3.2F8step 2.2

The ramification divisor. Define $R_{φ} = \sum_{p} l_{p} [p]$ with $l_{p}$ the length of the relative differentials at $p$ ; since $l_{p} = 0$ for all but finitely many $p$ and all $l_{p} \geq 0$ , this is an effective divisor on $P_{k}^{1}$ in the sense of [F8]. By step 2.2 its coefficients are $l_{0} = l_{\infty} = n - 1$ and $l_{p} = 0$ for every other closed point, so $R_{φ} = (n - 1) ([0] + [\infty])$ , with support ${0, \infty}$ and length $n - 1$ at each of the two points when $n \geq 2$ , and $R_{φ}$ is the zero divisor when $n = 1$ .

4.1F4F5F9step 1.1step 2.1step 3.1step 3.2∎

Conclusion. The morphism $φ ([s : t]) = [s^{n} : t^{n}]$ of the statement is finite and surjective of degree $n$ by steps 1.1 and 2.1; it is unramified at every closed point away from $0$ and $\infty$ by step 3.1; at $0$ and at $\infty$ the fibre is the single point with ramification index $n$ by steps 1.2 and 1.3, so the fibre degree sum reads $n = n$ over both points by step 2.1; and, in the tame case $char k ∤ n$ , the ramification divisor is $R_{φ} = (n - 1) ([0] + [\infty])$ by step 3.2. The Axiom of Choice of [F9] is used only through the construction of the finite morphism and through the divisor theory of the projective line, and no further selection is made.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-10-02Open item page →

Smooth plane quartic has genus three

Example

Let $k$ be algebraically closed and let $X = V_{+} (F) \subseteq P_{k}^{2}$ be a smooth plane quartic, so $\deg F = 4$ . Then $p_{a} (X) = \frac{(4 - 1) (4 - 2)}{2} = 3,$ every point of $X$ is regular so every delta invariant vanishes, and the geometric genus is $g (X) = 3$ . This realizes the triangular-number genus sequence $\frac{(d - 1) (d - 2)}{2}$ for smooth plane curves of degree $d$ .

Facts & Assumptions

Given: An algebraically closed field $k$ , a nonzero homogeneous form $F$ of degree $4$ , and the smooth plane hypersurface $X = V_{+} (F) \subseteq P_{k}^{2}$ . We work under the Axiom of Choice [A1].

[A1]

The Axiom of Choice is assumed (The Axiom of Choice).

[A2]

In ZF, AC implies Dependent Choice (AC implies DC implies countable choice). This supplies the DC use in the curve closed-subset finiteness route used by the delta-correction formula.

[F1]

If $X = V_{+} (F)$ is an integral plane curve of degree $d$ , then it is proper and has $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = 1 - χ (O_{X}) = \frac{(d - 1) (d - 2)}{2}$ . (Arithmetic genus of a plane curve)

[F11]

A curve over $k$ is nonempty, geometrically integral, separated, finite type, and of chain dimension one. (Curves over a field, Integral schemes)

[F2]

For an integral proper curve, the arithmetic genus is $p_{a} (X) = 1 - χ (O_{X})$ ; over algebraically closed $k$ , the geometric genus is $g (X) = g (X^{n u})$ , and the delta invariant vanishes exactly at regular points. (Genus and arithmetic genus of a curve, Geometric genus of a singular curve, Delta invariant of a curve singularity)

[F3]

For an integral plane curve over algebraically closed $k$ with isolated singularities, $g (X^{n u}) = \frac{(d - 1) (d - 2)}{2} - \sum_{x} δ_{x} (X),$ where the finite sum is over the singular points. The finiteness route uses DC under AC [A2] through the proper-closed-subset lemma for curves. (Geometric genus of a plane curve by delta invariants, Proper closed subsets of a curve are finite)

[F4]

Smoothness over $k$ makes every local ring regular. For a closed $k$ -point on an affine hypersurface chart with one actual equation in two variables and local dimension one, the Jacobian criterion says the local ring is regular exactly when the one-row Jacobian has rank one. (Smoothness over a field by geometric regularity, Jacobian rank detects regularity at closed points)

[F5]

A finite-variable polynomial ring over a field is a UFD, and every irreducible element is prime. (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes)

[F6]

Two positive-degree homogeneous forms in three variables with no common nonconstant factor have a nonempty finite projective intersection; over algebraically closed $k$ its closed points have residue field $k$ . (Algebraic Bezout formula as a sum of local scheme lengths)

[F7]

The standard charts of $P_{k}^{2}$ are affine planes and their pairwise overlaps are nonempty, so $P_{k}^{2}$ is irreducible; its projective dimension is two. A nonzero homogeneous form of positive degree cuts out a nonempty projective hypersurface whose irreducible components all have dimension one. (Relative projective space from standard charts, standard projective opens are affine spaces, Affine and projective n-space have dimension n, Nontrivial projective hypersurface sections)

[F8]

The equation on each standard affine chart of $V_{+} (F)$ is the dehomogenization of $F$ . At a closed point on a pure one-dimensional finite-type scheme over algebraically closed $k$ , the residue field is $k$ , and the affine local-dimension formula then gives local-ring dimension one. (projective hypersurface affine pieces, Finite-type maps from Jacobson rings induce finite residue-field extensions at maximal ideals, Local fibre dimension equals local ring dimension plus residue transcendence degree, Finite-variable polynomial algebras over fields are Noetherian by finite generators)

[F9]

Every regular local ring is normal. A normal integral curve is its own normalization by the normalization theorem's initiality. (regular local rings are normal, Weil divisor normal noetherian scheme, Normalization of an integral finite-type curve by gluing affine integral closures)

[F12]

If $S$ is a standard graded domain with $S_{+} \neq 0$ , then $(0)$ is a homogeneous prime not containing $S_{+}$ and is the generic point of $Proj S$ . Each nonempty standard chart ring is a degree-zero subring of a localization of $S$ at a nonzero homogeneous element, hence a domain; therefore $Proj S$ is integral. (Projective scheme of a homogeneous quotient and its standard affine charts, Integral schemes)

Verification

technique · establish that smoothness forces the quartic form to be square-free and irreducible, then compute the arithmetic and geometric genera

1.1F7F8

Dimension of the hypersurface. The nonzero quartic $F$ does not vanish identically on the irreducible surface $P_{k}^{2}$ . By [F7], $X = V_{+} (F)$ is nonempty and each of its irreducible components has dimension one. Thus every closed point used below has local dimension one by [F8].

2.1F4F5F7F8step 1.1

No repeated factor. Factor $F$ into irreducible homogeneous forms in the UFD $k [x_{0}, x_{1}, x_{2}]$ [F5]; homogeneous factors can be taken homogeneous because the lowest and highest graded degrees of a product add. Suppose an irreducible factor $G$ occurs with multiplicity at least two, so $F = G^{2} H$ . By [F7], $V_{+} (G)$ is nonempty; choose a closed point $P \in V_{+} (G)$ . In a standard affine chart through $P$ , the actual equation is $f = g^{2} h$ , so its first partial derivatives all vanish at $P$ . This remains true in every characteristic because each derivative is divisible by $g$ . By [F8] the local dimension is one, so [F4] says the zero Jacobian row makes the local ring nonregular. This contradicts smoothness. Thus $F$ is square-free.

3.1F4F5F6F8step 1.1step 2.1

No reducible square-free factorization. If square-free $F$ were reducible, write $F = G H$ with coprime homogeneous forms $G, H$ of positive degree. By [F6], their projective intersection is nonempty; choose a closed point $P$ in it. On a standard chart through $P$ , the equation is $f = g h$ with $g (P) = h (P) = 0$ , so every first partial derivative $h \partial g + g \partial h$ vanishes at $P$ . Its local ring has dimension one [F8] and is nonregular by [F4], contradicting smoothness again. Therefore $F$ is irreducible.

4.1F5F7F11F12step 1.1step 3.1

The curve hypothesis. By [F5], irreducible $F$ is prime, so the homogeneous coordinate ring $k [x_{0}, x_{1}, x_{2}] / (F)$ is a domain. The nonempty standard projective charts are spectra of domains, so [F12] makes its Proj reduced and irreducible; it is nonempty and one-dimensional by [F7]. Since $k$ is algebraically closed, its algebraic-closure fibre is itself, so $X$ is geometrically integral. As a closed subscheme of projective space, $X$ is separated and finite type. Hence $X$ is a curve over $k$ by [F11].

5.1A1A2F1F2F3F4step 4.1

The arithmetic genus and delta invariants. Smoothness makes every local ring regular [F4], so every delta invariant is zero [F2] and the sum in [F3] is empty. Applying [F1] with $d = 4$ gives $H^{0} (X, O_{X}) = k$ and $p_{a} (X) = \frac{(4 - 1) (4 - 2)}{2} = 3$ . The Axiom of Choice [A1] supplies the DC needed in [F3] through [A2].

6.1F1F2F3F9step 2.1step 3.1step 4.1step 5.1∎

The geometric genus. Applying [F3] to the integral quartic and using step 5.1 gives $g (X^{n u}) = 3$ . By [F9], smoothness makes $X$ normal, so its normalization is isomorphic to $X$ ; therefore $g (X) = g (X^{n u}) = 3$ , agreeing with $p_{a} (X)$ . More generally, the same square-free and irreducibility argument of steps 2.1 and 3.1 applies to any smooth plane curve of degree $d \geq 1$ over this algebraically closed field, so it is integral. Then [F1] gives $p_{a} = (d - 1) (d - 2) / 2$ , smoothness makes every delta invariant zero by [F2], and [F3] and [F9] give $g = p_{a}$ . Thus the displayed quartic is the $d = 4$ case of the triangular-number formula, without using a separate general-genus supplier.

Smooth Proper Curves Divisors Genus and Ramification — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Divisors and complete linear systems on the projective line

Example

Facts & Assumptions

Proof

A smooth conic is a projective line once it has a rational point

Example

Facts & Assumptions

Proof

Ramification of the double cover y^2=f(x)

Example

Facts & Assumptions

Verification

A torsion-only extension of the canonical formula fails for Frobenius

Statement refuted

Facts & Assumptions

Construction

Verification

Nodal cubic: arithmetic genus one, delta one, geometric genus zero

Example

Facts & Assumptions

Proof

Cuspidal cubic: delta invariant and normalization

Example

Facts & Assumptions

Proof

Smoothness of the source cannot be dropped in the extension of rational maps

Statement refuted

Facts & Assumptions

Refutation

Divisor degree with residue degrees over a nonclosed field

Example

Facts & Assumptions

Proof

A linear system with and without a base point

Example

Facts & Assumptions

Proof

A nontrivial degree-zero line bundle has no nonzero section

Counterexample

Facts & Assumptions

Proof

Ramification indices of the power map on the projective line

Example

Facts & Assumptions

Proof

Smooth plane quartic has genus three

Example

Facts & Assumptions

Verification

Sources