Residues Serre Duality for Curves and the Full Riemann Roch Theorem — Examples

Given: the Axiom of Choice, a perfect field $k$, $C={\mathbb{P}}_k^1$ with coordinate $t=x_1\mathrm{/}{x}_0$, a monic irreducible $g\in k[t]$, and the finite-support principal part $t^{-1}dt$ at the origin.

The Axiom of Choice is inherited from the cited cohomology, principal-parts, global-residue, and duality suppliers; the computations here make no additional choices. (The Axiom of Choice)

The standard charts are $U_0=\mathrm{Spec}k[t]$ and $U_1=\mathrm{Spec}k[s]$, with $s=1\mathrm{/}t$ on their overlap, and infinity has local parameter $s$. Closed points in $U_0$ correspond to monic irreducible polynomials $g$, with residue field $k[t]\mathrm{/}(g)$. (Relative projective space from standard charts, Divisors on the projective line are classified by degree, Degree divisor proper curve)

At $p=V(g)$, $u=g(t)$ is a uniformizer. Since $k$ is perfect, the irreducible polynomial $g$ is separable, so $g^{\mathrm{\prime }}(t)$ is a unit modulo $(g)$; the chain rule gives $du=g^{\mathrm{\prime }}(t)dt$. (Local rings at closed points of smooth curves are discrete valuation rings, Perfect fields: every irreducible polynomial is separable)

At a closed point with finite separable residue field, residue is the field trace of the coefficient of $u^{-1}du$ in a local parameter $u$; the residue is independent of the chosen parameter. (Residue of a rational differential at a separable closed point, The residue is independent of the uniformizer)

Formal Laurent-series residue extracts the coefficient of exponent $-1$. (Formal Laurent series $K((x))$, their order, derivative, and residue)

Over a perfect field, the sum of residues of a rational differential on a smooth proper geometrically integral curve is zero. (The global residue theorem on a smooth proper curve over a perfect field)

The principal-parts presentation represents $H^1(C,\omega )$ by finite-support local principal parts modulo rational principal parts. In particular, a single local Laurent tail at the rational origin gives a class. (H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections, Canonical bundle and canonical divisors)

For ${\mathbb{P}}_k^1$, $h^1(\omega )=h^0(\mathcal{O})=1$. The fixed normalized Gysin trace on $H^1(C,\omega )$ is the negative of the positive residue-sum functional over a perfect field. (h^1 of a line bundle equals the dimension of the space of dual sections, Normalization of the trace for Serre duality on a curve)

Given: the Axiom of Choice, a field $k$, ${\mathbb{P}}_k^1$ with coordinate $t=x_1\mathrm{/}{x}_0$, an integer $d\ge 0$, and $\mathcal{L}=\mathcal{O}(-d-2)$.

The Axiom of Choice is inherited from the projective cohomology, principal-parts, field-extension, and duality suppliers, and is used to take an algebraic closure in step 2.1. No other selection is made. (The Axiom of Choice)

On ${\mathbb{P}}_k^1$, $H^0(\mathcal{O}(d))$ has basis $x_0^{d-m}{x}_1^m$ for $0\le m\le d$, and $H^1(\mathcal{O}(-d-2))$ has Laurent basis $x_0^{e_0}{x}_1^{e_1}$ with $e_0,e_1<0$ and $e_0+e_1=-d-2$; both groups have dimension $d+1$. (Cohomology of O(d) on projective space)

The standard frames satisfy $x_0^{d+2}=s^{d+2}{x}_1^{d+2}$ on the overlap, $s=1\mathrm{/}t$, and $dt=-s^{-2}ds$. Also $dt$ maps to $x_0^{-2}$ and $ds$ maps to $-x_1^{-2}$ under ${\omega }_{{\mathbb{P}}^1}\cong \mathcal{O}(-2)$. Thus $t^{m}dt\otimes x_0^{d+2}=-s^{d-m}ds\otimes x_1^{d+2}$ and corresponds to $x_0^{d-m}{x}_1^m$. The sheaves $\mathcal{O}(r)$ are invertible and $\mathcal{O}(r{)}^{-1}\cong \mathcal{O}(-r)$. (Twisting sheaf on Proj, Relative projective space from standard charts, Invertible sheaves, Canonical bundle and canonical divisors)

Principal parts give the cokernel description of $H^1$; in particular, each finite-support tail $t^{-j}$ in the frame $x_0^{-d-2}$ represents the class $c_j$. (H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections)

At the rational origin with parameter $t$, the local residue of $t^{m-j}dt$ is its $t^{-1}$ coefficient. Over a perfect field, the positive residue pairing is the sum of these local residues. (Residue of a rational differential at a separable closed point, The residue pairing of a line bundle with the dual canonical twist)

For a smooth proper geometrically integral curve, the fixed normalized Gysin trace and its Serre pairing are defined over every field. Over a perfect field the trace pairing is the negative of the positive residue pairing. (Serre duality for line bundles on a smooth proper curve, and the residue realization, Normalization of the trace for Serre duality on a curve)

For a proper scheme, coherent cohomology commutes with arbitrary field extension. The fixed Gysin trace and its cup/evaluation pairing also commute with extension of the base field. (Flat field extension commutes with coherent cohomology, Embedding compatibility of smooth-projective Gysin traces)

On ${\mathbb{P}}_k^1$, ${\omega }_{{\mathbb{P}}^1}\cong \mathcal{O}(-2)$: its canonical degree is $-2$, and the Picard group is classified by degree. (The canonical divisor has degree 2g - 2, The Picard group of the projective line, Divisors on the projective line are classified by degree, Degree divisor proper curve, Canonical bundle and canonical divisors)

Given: the Axiom of Choice and its consequence Dependent Choice; a field $k$, the projective line $C={\mathbb{P}}_k^1$ with point at infinity $\mathrm{\infty }$, and a divisor $D$ of degree $d={\mathrm{deg}}_{k}D$.

Every divisor on ${\mathbb{P}}_k^1$ is linearly equivalent to $({\mathrm{deg}}_{k}D)[\mathrm{\infty }]$, and the associated invertible sheaf of $[\mathrm{\infty }]$ is $\mathcal{O}(1)$; the divisor and invertible-sheaf dictionaries agree on ${\mathbb{P}}_k^1$. (Divisors on the projective line are classified by degree, Cartier and Weil divisors agree on a smooth curve, Invertible sheaf of cartier divisor)

$\mathrm{Pic}({\mathbb{P}}_k^1)\cong \mathbb{Z}$, the class of $\mathcal{O}(1)$ corresponding to $1$; hence $\mathcal{O}(D)\cong \mathcal{O}(d)$ for $D$ of degree $d$, and $\mathrm{\ell }(D)=h^0(C,\mathcal{O}(D))$ is the dimension of the Riemann-Roch space of $D$. (The Picard group of the projective line, The Riemann-Roch dimension l(D))

On ${\mathbb{P}}_k^1$, $H^0({\mathbb{P}}_k^1,\mathcal{O}(m))$ is the degree-$m$ part of $k[x_0,x_1]$, of dimension $m+1$ for $m\ge 0$, and is $0$ for $m<0$; the twisting sheaves $\mathcal{O}(m)$ are the ones attached to the standard charts. (Cohomology of O(d) on projective space, Twisting sheaf on Proj)

For a smooth proper geometrically integral curve of genus $g$, the canonical divisor has degree $2g-2$; on ${\mathbb{P}}_k^1$ the genus is $0$, so ${\mathrm{deg}}_k{K}_C=-2$, and $\mathcal{O}(K_C)={\omega }_C$ with $K_C\sim -2[\mathrm{\infty }]$ because every degree-$(-2)$ divisor is linearly equivalent to $-2[\mathrm{\infty }]$. (The canonical divisor has degree 2g - 2, Canonical bundle and canonical divisors, Degree divisor proper curve, [F1])

The full Riemann-Roch theorem states $\mathrm{\ell }(D)-\mathrm{\ell }(K_C-D)={\mathrm{deg}}_k(D)+1-g$ for a divisor $D$ on a smooth proper geometrically integral curve of genus $g$ over an arbitrary field, and the index of speciality is $i(D)=\mathrm{\ell }(K_C-D)$, with $D$ nonspecial when $i(D)=0$. (The full Riemann-Roch theorem for divisors on a smooth proper curve, The index of speciality i(D), Special and nonspecial divisors)

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

In ZF, the Axiom of Choice implies Dependent Choice; this supplies the Dependent Choice premise of the Cartier-to-Weil divisor dictionary used in [F1] and [F4]. (AC implies DC implies countable choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Given: the Axiom of Choice and its consequence Dependent Choice; a smooth proper geometrically integral curve $C$ of genus one over a field $k$, an invertible sheaf $\mathcal{L}$ of degree $n\ge 1$, and the associated divisor $D$.

For an invertible sheaf of degree $>2g-2$ on a smooth proper geometrically integral curve of genus $g$, $H^1(C,\mathcal{L})=0$ and $h^0(C,\mathcal{L})=\mathrm{deg}\mathcal{L}+1-g$; equivalently $\mathrm{\ell }(D)={\mathrm{deg}}_{k}D+1-g$ for divisors of degree $>2g-2$. (H^1 of a line bundle vanishes above degree 2g - 2, Riemann-Roch in exact form for divisors of degree above 2g - 2)

The full Riemann-Roch theorem reads $\mathrm{\ell }(D)-\mathrm{\ell }(K_C-D)={\mathrm{deg}}_{k}D+1-g$; the line bundle associated with $D$ has $h^0(C,\mathcal{O}(D))=\mathrm{\ell }(D)$, and the degree of ${\mathcal{L}}^{-1}$ is $-\mathrm{deg}\mathcal{L}$. (The full Riemann-Roch theorem for divisors on a smooth proper curve, The Riemann-Roch dimension l(D), Invertible sheaf of cartier divisor, Cartier and Weil divisors agree on a smooth curve)

On a genus-one curve ${\omega }_C\cong {\mathcal{O}}_C$, and Serre duality for invertible sheaves gives $h^1(C,\mathcal{L})=h^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})$; a line bundle of degree $<0$ has no nonzero global section. (The canonical bundle of a genus-one curve is trivial, Serre duality for line bundles on a smooth proper curve, and the residue realization, Negative-degree line bundles have no nonzero sections)

The dimension of the complete linear system of a divisor is $\dim \mathrm{\mid }D\mathrm{\mid }=\mathrm{\ell }(D)-1={\mathrm{deg}}_{k}D-g+i(D)$, where $i(D)=\mathrm{\ell }(K_C-D)$ is the index of speciality. Under an isomorphism $\mathcal{L}\cong {\mathcal{O}}_C(D)$, a nonzero global section corresponds to a nonzero $f\in L(D)$; its zero divisor is the effective divisor $D+\mathrm{div}(f)$, of degree ${\mathrm{deg}}_{k}D$ because principal divisors have degree zero. (The dimension of a complete linear system, Complete linear system, The space L(D), Degree divisor proper curve, Rational sections of line bundles are Cartier divisors, Principal divisors on a normal proper curve have degree zero)

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

In ZF, the Axiom of Choice implies Dependent Choice; this supplies the Dependent Choice premise of the Cartier-to-Weil dictionary used in [F2] and [F4]. (AC implies DC implies countable choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Given: the Axiom of Choice and its consequence Dependent Choice; a field $k$ and a smooth plane cubic $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ of degree $d=3$.

For a smooth plane curve $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ of degree $d$, adjunction gives ${\omega }_C\cong {\mathcal{O}}_C(d-3)$, and the genus is $(d-1)(d-2)\mathrm{/}2$; the degree of the degree-$d$ hypersurface is $\mathrm{deg}F=d$. (Adjunction for smooth plane curves, The genus of a smooth plane curve in terms of its degree, degree projective hypersurface)

For a smooth proper geometrically integral curve of genus $g$, ${\mathrm{deg}}_k{\omega }_C=2g-2$ and $h^0(C,{\omega }_C)=g$, with ${\omega }_C={\mathcal{O}}_C(K_C)$ the canonical bundle. (The canonical divisor has degree 2g - 2, The canonical bundle has exactly g independent sections, Canonical bundle and canonical divisors, Degree divisor proper curve)

An invertible sheaf of degree $0$ on a smooth proper curve that has a nonzero global section is trivial; equivalently, an effective divisor of degree $0$ is $0$, so a degree-zero divisor whose sheaf has a section is principal. (A degree-zero line bundle with a nonzero section is trivial, Rational sections of line bundles are Cartier divisors, Invertible sheaves)

A genus-one curve over $k$ with a $k$-rational point $p_0$ embeds as a smooth plane cubic via the degree-three very ample invertible sheaf ${\mathcal{O}}_C(3p_0)$; and on a genus-one curve the canonical bundle is trivial. (A genus-one curve with a rational point embeds as a plane cubic, The canonical bundle of a genus-one curve is trivial)

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

In ZF, the Axiom of Choice implies Dependent Choice; this supplies the Dependent Choice premise of the Cartier-to-Weil dictionary used in [F3] and the cited genus-one and degree suppliers. (AC implies DC implies countable choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Given: AC, a field $k$ and a smooth plane quartic $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ of degree $d=4$.

For a smooth plane curve $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ of degree $d$, ${\omega }_C\cong {\mathcal{O}}_C(d-3)$ and $g(C)=(d-1)(d-2)\mathrm{/}2$; the degree of the hypersurface is $\mathrm{deg}F=d$. (Adjunction for smooth plane curves, The genus of a smooth plane curve in terms of its degree, degree projective hypersurface)

For a smooth proper geometrically integral curve of genus $g$, ${\mathrm{deg}}_k{\omega }_C=2g-2$ and $h^0(C,{\omega }_C)=g$, and, when the complete canonical system $\mathrm{\mid }{K}_C\mathrm{\mid }$ is base-point-free and its section space is nonzero, it defines the canonical morphism (The canonical divisor has degree 2g - 2, The canonical bundle has exactly g independent sections, Canonical bundle and canonical divisors, Complete linear system, A base-point-free linear system defines a morphism to projective space).

On ${\mathbb{P}}_k^2$ one has $H^1({\mathbb{P}}^2,\mathcal{O}(m))=0$ for every $m$, $H^0({\mathbb{P}}^2,\mathcal{O}(-3))=0$, and $H^0({\mathbb{P}}^2,\mathcal{O}(1))$ is the space of linear forms; the twisting sheaves are the ones attached to the standard graded presentation. (Cohomology of O(d) on projective space, Twisting sheaf on Proj)

If $i:C\hookrightarrow {\mathbb{P}}_k^2$ is the plane quartic, the published hypersurface sequence gives $0\to {\mathcal{O}}_{{\mathbb{P}}^2}(-4)\stackrel{\cdot F}{\to }{\mathcal{O}}_{{\mathbb{P}}^2}\to i_{\ast }{\mathcal{O}}_C\to 0$: the nonzero dehomogenizations of $F$ are nonzerodivisors in the polynomial domain rings of the standard charts. The standard graded polynomial ring generated in degree one makes ${\mathcal{O}}_{{\mathbb{P}}^2}(1)$ invertible, so tensoring preserves exactness. On each standard chart the quotient by the local equation $F$ with the restricted twist identifies the last term with $i_{\ast }{\mathcal{O}}_C(1)$, yielding $0\to {\mathcal{O}}_{{\mathbb{P}}^2}(-3)\stackrel{\cdot F}{\to }{\mathcal{O}}_{{\mathbb{P}}^2}(1)\to i_{\ast }{\mathcal{O}}_C(1)\to 0$. (Hypersurface cohomology sequence, Invertible twists for degree-one generated rings, Relative projective space from standard charts, Closed immersions are affine quotients and survive base change, Direct image of a sheaf along a continuous map, Twisting sheaf on Proj)

The short exact sequence in [F4] gives a long exact sequence in sheaf cohomology, and closed-immersion pushforward identifies $H^q({\mathbb{P}}^2,i_{\ast }{\mathcal{O}}_C(1))$ with $H^q(C,{\mathcal{O}}_C(1))$. In particular its low-degree segment is $0\to H^0({\mathbb{P}}^2,\mathcal{O}(-3))\to H^0({\mathbb{P}}^2,\mathcal{O}(1))\to H^0(C,{\mathcal{O}}_C(1))\to H^1({\mathbb{P}}^2,\mathcal{O}(-3))$. (Long exact sequence of sheaf cohomology, Closed immersion preserves cohomology and coherent pushforward)

For a smooth proper geometrically integral curve of genus $g\ge 2$, the canonical map is a closed immersion exactly when $g\ge 3$ and the curve is geometrically nonhyperelliptic, meaning that its algebraic-closure base change has no degree-two map to ${\mathbb{P}}^1$. (The canonical map: base-point-freeness and the hyperelliptic exception, Hyperelliptic curves and hyperelliptic maps)

For a divisor $D$ on a smooth proper curve, the complete linear system $\mathrm{\mid }D\mathrm{\mid }$ is the set of effective divisors linearly equivalent to $D$, and its dimension is $\mathrm{\ell }(D)-1$. (Complete linear system, Degree divisor proper curve)