Riemann Roch for Curves via Euler Characteristics — Examples

Given: the Axiom of Choice inherited from the current projective-line cohomology and divisor/Picard suppliers; a field $k$, the projective line ${\mathbb{P}}_k^1$ with coordinate $t$ and point at infinity $\mathrm{\infty }$, and the divisor $D=d[\mathrm{\infty }]$ for an integer $d$.

Projective-line data: ${\mathbb{P}}_k^1$ is a smooth proper geometrically integral curve over $k$ of genus $0$; the coordinate section $x_0$ of $\mathcal{O}(1)$ vanishes exactly at infinity with multiplicity one, so $\mathrm{div}(x_0)=[\mathrm{\infty }]$ and $\mathcal{O}(1)\cong \mathcal{O}(\mathrm{\infty })$ with ${\mathrm{deg}}_k\mathcal{O}(1)=1$; every divisor $D^{\mathrm{\prime }}$ is linearly equivalent to ${\mathrm{deg}}_k(D^{\mathrm{\prime }})[\mathrm{\infty }]$, and the degree homomorphism on divisor classes is an isomorphism, so linearly equivalent divisors have equal degree and $[\kappa (\mathrm{\infty }):k]=1$ (Divisors on the projective line are classified by degree, The Picard group of the projective line).

Global sections: for $n=1$ and $A=k$ one has $H^0({\mathbb{P}}_k^1,\mathcal{O}(d))\cong k[x_0,x_1{]}_d$ for $d\ge 0$ and $H^0({\mathbb{P}}_k^1,\mathcal{O}(d))=0$ for $d<0$; the monomials $x_0^a{x}_1^{d-a}$ for $0\le a\le d$ form a $k$-basis of $k[x_0,x_1{]}_d$, so $h^0(\mathcal{O}(d))=d+1$ for $d\ge 0$ and $h^0(\mathcal{O}(d))=0$ for $d<0$ (Global sections of projective twists).

Top cohomology: for $n=1$ the group $H^1({\mathbb{P}}_k^1,\mathcal{O}(d))$ vanishes for $d>-2$ and for $d\le -2$ it is free of rank $\left(\genfrac{}{}{0px}{}{-d-1}{1}\right)=-d-1$ over $k\mathrm{\ne }0$; so $h^1(\mathcal{O}(d))=0$ for $d\ge -1$ and $h^1(\mathcal{O}(d))=-d-1$ for $d\le -2$ (Top cohomology of projective twists).

Degree: for a divisor $D={\sum }_x{n}_x[x]$ the $k$-degree is ${\mathrm{deg}}_k(D)={\sum }_x{n}_x[\kappa (x):k]$, sum over the finite support, and it is a group homomorphism on the divisor group (Degree divisor proper curve, Divisors on a smooth proper curve).

Riemann-Roch dimensions: $l(D)={\dim }_{k}L(D)=h^0(D)={\dim }_k{H}^0(C,{\mathcal{O}}_C(D))$ and $i(D)=h^1(D)={\dim }_k{H}^1(C,{\mathcal{O}}_C(D))$, so $l(D)=h^0(\mathcal{O}(D))$ and $i(D)=h^1(\mathcal{O}(D))$ for the attached sheaf (The Riemann-Roch dimension l(D), The index of speciality i(D)).

Riemann-Roch: $l(D)-i(D)={\mathrm{deg}}_k(D)+1-g$ with $i(D)\ge 0$, and $l(D)={\mathrm{deg}}_k(D)+1-g$ exactly for the nonspecial divisors (Riemann-Roch as l minus i, Special and nonspecial divisors).

Nonspeciality: $D$ is nonspecial exactly when $i(D)=0$, and special exactly when $i(D)\ge 1$; equivalently nonspeciality is the equality case $l(D)={\mathrm{deg}}_k(D)+1-g$ (Special and nonspecial divisors).

The complete linear system $\mathrm{\mid }D\mathrm{\mid }=\{D^{\mathrm{\prime }}\text{ effective}:D^{\mathrm{\prime }}\text{ is linearly equivalent to }D\}$ is in bijection with the set of $k$-lines in $L(D)$ and is empty exactly when $L(D)=0$ (Complete linear system).

The current Divisors on the projective line are classified by degree and The Picard group of the projective line identify the divisor class and attached twist; The Riemann-Roch dimension l(D), The index of speciality i(D), Special and nonspecial divisors and Complete linear system supply the dimension and linear-system interpretations.

The Axiom of Choice is available and is inherited only through the suppliers named above; the computations below evaluate explicit formulas and select nothing (The Axiom of Choice).

Given: the Axiom of Choice inherited from the current plane-genus, rational-point and cohomology suppliers; a field $k$ of characteristic not two, a smooth plane conic curve $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ with arithmetic genus computed by the plane-curve theorem, and a $k$-rational point $p\in C$.

Divisors and degree: ${\mathrm{deg}}_k(D)={\sum }_x{n}_x[\kappa (x):k]$ is a group homomorphism, and a $k$-rational point has residue degree one, so ${\mathrm{deg}}_k[p]=1$ (Degree divisor proper curve).

Plane conic arithmetic genus: a curve $X=V_{+}(G)$ cut out by a nonzero homogeneous form of degree $d\ge 1$ has $H^0(X,{\mathcal{O}}_X)=k$ and $p_a(X)=1-\chi ({\mathcal{O}}_X)=\frac{(d-1)(d-2)}{2}$; for $d=2$ this is $0$, and for a smooth curve the arithmetic genus is the genus (Arithmetic genus of a plane curve, Curves over a field; the genus $g(C)=1-\chi (C,{\mathcal{O}}_C)$ is the one of Genus via the Euler characteristic, where the agreement with the arithmetic genus in the smooth case is recorded).

Rational-point theorem: a smooth proper geometrically integral curve of genus $0$ over $k$ that admits a divisor of degree one is isomorphic to ${\mathbb{P}}_k^1$ (A genus-zero curve with a degree-one divisor is the projective line).

Cohomology of the twists on the projective line: ${\mathbb{P}}_k^1$ is a smooth proper geometrically integral curve of genus $0$; for a $k$-rational point $q$ the degree-one divisor $[q]$ is linearly equivalent to $[\mathrm{\infty }]$, so $\mathcal{O}([q])\cong \mathcal{O}(1)$ with ${\mathrm{deg}}_k\mathcal{O}(1)=1$, while $H^0({\mathbb{P}}_k^1,\mathcal{O}(1))\cong k[x_0,x_1{]}_1$ has dimension $2$; hence $l([q])=h^0(\mathcal{O}(1))=2$ (Global sections of projective twists, Divisors on the projective line are classified by degree).

Riemann-Roch and the index of speciality: $l(D)-i(D)={\mathrm{deg}}_k(D)+1-g$ with $i(D)=h^1(D)\ge 0$; $i(D)=0$ if and only if $D$ is nonspecial, equivalently if and only if the identity $l(D)={\mathrm{deg}}_k(D)+1-g$ is an equality (Riemann-Roch as l minus i, The index of speciality i(D), Special and nonspecial divisors, The Riemann-Roch dimension l(D)).

Invariance under isomorphism: $l(D)={\dim }_k{H}^0(C,{\mathcal{O}}_C(D))$ and $i(D)={\dim }_k{H}^1(C,{\mathcal{O}}_C(D))$ are dimensions of cohomology groups of the attached invertible sheaf, so an isomorphism of curves carrying $D$ to a divisor $D^{\mathrm{\prime }}$ carries ${\mathcal{O}}_C(D)$ to ${\mathcal{O}}_{C^{\mathrm{\prime }}}(D^{\mathrm{\prime }})$ and preserves $l$ and $i$ (The Riemann-Roch dimension l(D), Sheaf cohomology as right derived global sections, The index of speciality i(D)).

The Axiom of Choice is available and is inherited only through the rational-point and Riemann-Roch suppliers above; the computation evaluates the given conic, point and isomorphism and selects nothing beyond them (The Axiom of Choice).

Given: the Axiom of Choice inherited from the current smoothness, properness, plane-genus and rational-point suppliers; the field $k=\mathbb{R}$, the form $F=x^2+y^2+z^2$, the closed subscheme $C=V_{+}(F)\subseteq {\mathbb{P}}_{\mathbb{R}}^2$, and its base change $C_{\mathbb{C}}$ to $\mathbb{C}$.

Smoothness: for either $K=\mathbb{R}$ or $K=\mathbb{C}$, each of the three standard affine charts of $C_K$ is presented as $A_K=K[u,v]\mathrm{/}(1+u^2+v^2)$. Its Jacobian row is $(2u,2v)$, and in $A_K$ one has $1=(-u\mathrm{/}2)(2u)+(-v\mathrm{/}2)(2v)$. Thus no prime contains both entries; around every prime one of them is invertible, so the one-equation presentation is standard smooth there by Standard smooth presentations and locally standard smooth maps. By [F2] the structure map $C_K\to \mathrm{Spec}K$ is proper, hence of finite type by Proper morphisms, so the finite-type hypothesis in Smooth morphisms via local standard smooth presentations holds. Therefore every chart, and hence $C_K$, is smooth over $K$.

Properness: for either $K=\mathbb{R}$ or $K=\mathbb{C}$, ${\mathbb{P}}_K^2$ is proper over $K$; the closed immersion $C_K\hookrightarrow {\mathbb{P}}_K^2$ is proper, and its composition with the structure morphism is proper (Finite-dimensional projective space is proper over every base, Closed immersions are proper, Properness survives composition, Proper morphisms).

Geometric integrality: over $\mathbb{C}$ the form $F$ has no factorisation into linear forms. Any proper factorisation of a homogeneous quadratic over a field has two degree-one factors; writing each as its degree-one homogeneous part plus a constant, the degree-zero and degree-one parts of their product force both constants to vanish. Thus it suffices to test homogeneous linear factors. If $F=L_1{L}_2$, then restricting to $z=0$ and using unique factorisation in $\mathbb{C}[x,y]$ lets us rescale and order the factors as $L_1=x+iy+\alpha z$ and $L_2=x-iy+\gamma z$. Their product has $xz$, $yz$, and $z^2$ coefficients $\alpha +\gamma$, $i(\gamma -\alpha )$, and $\alpha \gamma$, respectively. Equality with $F$ requires $\alpha +\gamma =0$, $i(\gamma -\alpha )=0$, and $\alpha \gamma =1$; the first two force $\alpha =\gamma =0$ in characteristic zero, contradicting the third. Thus $F$ is irreducible in $\mathbb{C}[x,y,z]$; its principal ideal is prime by the finite-variable UFD lemma, so $C_{\mathbb{C}}$ is reduced and irreducible. It is nonempty since $[i:0:1]\in C_{\mathbb{C}}$. The extension $\mathbb{C}\mathrm{/}\mathbb{R}$ is algebraic ($\mathbb{C}\mathrm{/}\mathbb{R}$ has power basis $1,i$ and degree $2$) and $\mathbb{C}$ is algebraically closed (The complex numbers are algebraically closed), so $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$ (An algebraic closure of a field). Hence $C_{\mathbb{C}}$ is the geometric fibre defining geometric integrality of $C$ over $\mathbb{R}$; for $C_{\mathbb{C}}$ over $\mathbb{C}$, take the algebraic closure to be $\mathbb{C}$ itself. The fibres are integral in the sense of Geometric properties of fibres, giving the geometric-integrality assertions in Curves over a field (Geometric fibres and geometric points, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes).

Scheme chain dimension: for $K\in \{\mathbb{R},\mathbb{C}\}$, the three standard affine charts of $C_K$ have coordinate ring $A_K=K[u,v]\mathrm{/}(1+u^2+v^2)$ (Relative projective space from standard charts). By [F3], $F$ is irreducible over $\mathbb{C}$ and therefore over $\mathbb{R}$; for either field $K$ its homogeneous ideal is prime by the UFD property of $K[x,y,z]$ (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes). Each chart ring is the degree-zero subring of the localization of this homogeneous domain at one coordinate, hence is a nonzero domain. The map $K[u]\to A_K$ is injective: if a nonzero $q(u)$ lay in $(1+u^2+v^2)$, then in $K[u][v]$ it would equal $(1+u^2+v^2)h$ for nonzero $h$, contradicting additivity of degree in $v$ (Over an integral domain, degrees add under multiplication of nonzero polynomials). Thus $u$ is transcendental over $K$, while $v$ is algebraic over $K(u)$ by $v^2+u^2+1=0$; the transcendence-degree tower formula gives ${\mathrm{trdeg}}_K\mathrm{Frac}(A_K)=1$ (Transcendence degree is additive in finite towers). The affine-domain dimension theorem gives Krull dimension $\dim A_K=1$ (Affine-domain dimension equals transcendence degree). The ring $A_K$ is Noetherian as a quotient of a finite-variable polynomial ring, so its spectrum is Noetherian (Finite-variable polynomial algebras over fields are Noetherian by finite generators, The spectrum of a Noetherian ring is a Noetherian topological space). Every nonempty irreducible closed subset of this spectrum has a prime ideal as its unique generic point, and distinct primes have distinct closures; thus chains of nonempty irreducible closed subsets have exactly the lengths of chains of prime ideals. The chart's chain dimension is therefore its Krull dimension, one (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). Each of the three charts is Noetherian. A descending chain of closed subsets of $C_K$ stabilizes after restriction to each chart; since the cover is finite, the maximum of those three stabilization indices works on all of $C_K$. Thus $C_K$ is Noetherian, and the open-cover dimension lemma gives its chain dimension as the supremum of the three chart dimensions, namely one (Dimension can be computed on an open cover).

Arithmetic genus: a curve $X=V_{+}(G)$ cut out by a nonzero homogeneous form of degree $d\ge 1$ has $p_a(X)=\frac{(d-1)(d-2)}{2}$; for $d=2$ this is $0$, and for a smooth curve $p_a=g$ (Arithmetic genus of a plane curve, Genus via the Euler characteristic).

Residue-field degrees: if $p$ is a closed point, an affine neighborhood $\mathrm{Spec}A$ of $p$ is of finite type over $\mathbb{R}$, and $p$ remains closed there, so $\kappa (p)=A\mathrm{/}{\mathfrak{m}}_p$ is finite over $\mathbb{R}$ (Curves over a field, A maximal ideal of an affine algebra has finite residue field over the base field). The field $\mathbb{R}$ is perfect because it has characteristic zero (Fields of characteristic zero, finite fields, and algebraically closed fields are perfect), so the finite extension $\kappa (p)\mathrm{/}\mathbb{R}$ is separable (Every algebraic extension of a perfect field is separable) and simple (A finite extension generated by elements all but possibly one of which are separable is simple). If $\kappa (p)=\mathbb{R}(\alpha )$, the minimal polynomial of $\alpha$ is irreducible and has degree $[\kappa (p):\mathbb{R}]$ (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element, A simple algebraic extension is its minimal-polynomial quotient and has power basis $1,a,\dots ,a^{n-1}$ and degree $n$); every irreducible real polynomial has degree $1$ or $2$ (An irreducible polynomial in $\mathbb{R}[x]$ has degree $1$ or $2$). If this degree were $1$, then $\kappa (p)=\mathbb{R}$ and the residue-field point would give an $\mathbb{R}$-morphism $\mathrm{Spec}\mathbb{R}\to C$ (Field-valued points and local-ring points).

No real points and even divisor degrees: every real sum of squares $x^2+y^2+z^2$ vanishes only at the origin, so $C(\mathbb{R})=\mathrm{\varnothing }$. Therefore [F6] excludes residue degree $1$ for every closed point, and every closed point has residue degree $2$. A divisor is a finite signed combination $D={\sum }_p{n}_p[p]$ of closed points and has degree ${\mathrm{deg}}_{\mathbb{R}}D={\sum }_p{n}_p[\kappa (p):\mathbb{R}]$ (Degree divisor proper curve, The residue field at a point of an affine scheme); hence ${\mathrm{deg}}_{\mathbb{R}}D=2{\sum }_p{n}_p$ is even.

The rational-point theorem: a smooth proper geometrically integral curve of genus $0$ over a field $K$ that admits a divisor of degree one (equivalently a $K$-rational closed point) is isomorphic to ${\mathbb{P}}_K^1$ (A genus-zero curve with a degree-one divisor is the projective line); and an isomorphism of $K$-schemes induces a bijection of $K$-rational points, while ${\mathbb{P}}_K^1$ has the $K$-point $[1:0]$.

The Axiom of Choice is assumed in the local-standard-smooth definition used in [F1], in the three properness results used in [F2], in the Noetherian-spectrum and irreducible-closed-subset correspondences used in [F4], and in the genus-zero rational-point theorem [F8]; the arithmetic-genus route in [F5] also inherits the properness suppliers. The factorisation, chart-dimension computations, and residue-field degree argument require no further choice principle, and the algebraic closure used here is the explicitly given $\mathbb{C}$ (The Axiom of Choice).