Surface Riemann-Roch and the Hodge Index Theorem

Given: a field $k$, an integral smooth projective surface $X$ over $k$, an ample invertible ${\mathcal{O}}_X$-module $H$, a nonzero effective Cartier divisor $D\subseteq X$, and (for part 3) an invertible sheaf $\mathcal{M}$ with a nonzero global section $s$.

$X$ is projective over $k$ in the H-projective convention, hence proper over $k$ (Projective morphisms are proper) and of finite type, so it is Noetherian and locally Noetherian; coherent ${\mathcal{O}}_X$-modules have finite-dimensional cohomology and a well-defined Euler characteristic $\chi$ (Euler characteristic of a coherent sheaf, Intersection numbers of Cartier divisors on a smooth projective surface). The intersection product on invertible sheaves and Cartier divisors is symmetric and $\mathbb{Z}$-bilinear, vanishes against ${\mathcal{O}}_X$, and depends only on the isomorphism classes of the entries (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear).

For an effective Cartier divisor $C$ on $X$ and any Cartier divisor $E$ one has $C\cdot E={\mathrm{deg}}_C({\mathcal{O}}_X(E){\mathrm{\mid }}_C)$, the degree being ${\mathrm{deg}}_C(\mathcal{N})=\chi (C,\mathcal{N})-\chi (C,{\mathcal{O}}_C)$ of Degree of an invertible sheaf on a proper one-dimensional scheme on the proper curve $C$ (Intersection with a curve is the degree of the restriction).

A nonzero effective Cartier divisor $D$ on the surface $X$ is nonempty and has pure dimension one. Nonemptiness: the vanishing subscheme is empty exactly for the zero divisor (Effective cartier divisor). Dimension: on an affine chart $U=\mathrm{Spec}A$ meeting $\mathrm{Supp}D$, the coordinate ring $A$ is a finite-type $k$-domain of dimension two, and at a closed point $x\in U$ with maximal ideal ${\mathfrak{m}}_x$ one has $\mathrm{ht}({\mathfrak{m}}_x)=\dim A=2$ (Maximal ideals of an affine domain have full height); the local equation $f_x$ of $D$ at $x$ is a nonzerodivisor and a nonunit, so every prime minimal over $(f_x)$ has height one (A minimal prime over a principal nonzerodivisor has height one) and $\dim A\mathrm{/}\mathfrak{p}=\dim A-\mathrm{ht}(\mathfrak{p})=1$ (Height plus quotient dimension equals ambient dimension in an affine domain); hence $\mathrm{Supp}D$ has dimension one, and it has dimension at most one everywhere by the same height computation on the remaining charts (Chain dimension and the empty-space convention, A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point). Thus $D$ is a proper $k$-scheme of pure dimension one (Projective morphisms are proper), and for the very ample embedding used below the coherent sheaf $\mathcal{F}=i_{\ast }{\mathcal{O}}_D$ is nonzero with $\dim \mathrm{Supp}\mathcal{F}=1$ (Integral schemes).

Very ample powers: since $\mathrm{Spec}k$ is Noetherian, $X\to \mathrm{Spec}k$ is proper of finite type and $H$ is ample, there is an integer $m\ge 1$ such that $H^{\otimes m}$ is closed H-very ample relative to $\mathrm{Spec}k$ (High powers of an ample line bundle embed a proper scheme); by definition this means that for some $N\ge 0$ there is a closed immersion $i:X\hookrightarrow {\mathbb{P}}_k^N$ with $i^{\ast }{\mathcal{O}}_{{\mathbb{P}}^N}(1)\cong H^{\otimes m}$ (Relative very ampleness in the finite projective-space convention). Fix such $m,i,N$ and write ${\mathcal{O}}_X(1):=H^{\otimes m}$ for the embedding $i$.

Hilbert polynomial: for a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ and the twist $\mathcal{F}(n)=\mathcal{F}\otimes {\mathcal{O}}_X(1{)}^{\otimes n}$ the function $n\mapsto \chi (X,\mathcal{F}(n))$ is a polynomial $P_{\mathcal{F}}$ of degree $\dim \mathrm{Supp}\mathcal{F}$, with exact degree and nonzero leading coefficient when $\mathcal{F}\mathrm{\ne }0$, and $P_{\mathcal{F}}(n)=h^0(X,\mathcal{F}(n))$ for $n\gg 0$ (Hilbert function and Euler characteristic on a projective scheme, Euler characteristic is a Hilbert polynomial, Degree of the coherent Hilbert polynomial). Moreover for $n\gg 0$ all higher cohomology of $\mathcal{F}(n)$ vanishes (Serre vanishing for coherent sheaves and ample twists).

For a closed immersion $j:D\hookrightarrow X$ and an invertible sheaf $A$ on $X$, the projection formula gives $A^{\otimes n}\otimes j_{\ast }{\mathcal{O}}_D\cong j_{\ast }(A^{\otimes n}{\mathrm{\mid }}_D)$ and $\chi (X,j_{\ast }(A^{\otimes n}{\mathrm{\mid }}_D))=\chi (D,A^{\otimes n}{\mathrm{\mid }}_D)$ for every integer $n$ (Projection formula for a closed immersion and an invertible sheaf).

A nonzero global section of an invertible sheaf on the integral scheme $X$ is a regular section, its zero scheme $Z(s)$ is an effective Cartier divisor with ${\mathcal{O}}_X(Z(s))\cong \mathcal{M}$ and $Z(s)=\mathrm{\varnothing }$ exactly when $s$ is nowhere vanishing, in which case $\mathcal{M}\cong {\mathcal{O}}_X$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Zero scheme of a line-bundle section).

The Axiom of Choice is inherited from the cohomology, Hilbert-polynomial and ample-embedding suppliers of [F4]–[F6]; the divisor $D$, the section $s$ and the embedding $i$ are given data, and only finitely many sheaves and Hilbert-polynomial values are used below.

Given: a field $k$, an integral smooth projective surface $X$ over $k$, an ample invertible sheaf $L$ and an invertible sheaf $M$ on $X$.

Serre's global-generation criterion: on the Noetherian scheme $X$ the invertible sheaf $L$ is ample if and only if for every coherent ${\mathcal{O}}_X$-module $F$ the twist $F\otimes L^{\otimes n}$ is globally generated for all sufficiently large $n$ (Serre global-generation criterion for ampleness, Locally Noetherian and Noetherian schemes). The invertible sheaf $M$ is coherent on the locally Noetherian scheme $X$, and $X$ is quasi-compact, so global generation of an invertible sheaf is witnessed by finitely many global sections (Global generation by the evaluation map, Invertible sheaves).

Ample powers embed: $\mathrm{Spec}k$ is Noetherian, $X\to \mathrm{Spec}k$ is proper of finite type, and $L$ is ample, so there is an integer $d_0\ge 1$ such that $L^{\otimes d}$ is closed H-very ample relative to $\mathrm{Spec}k$ for every $d\ge d_0$ (High powers of an ample line bundle embed a proper scheme, Projective morphisms before Proj). Concretely this means that for each such $d$ there is a closed immersion $i_d:X\hookrightarrow {\mathbb{P}}_k^{N_d}$ with $i_d^{\ast }\mathcal{O}(1)\cong L^{\otimes d}$ (Relative very ampleness in the finite projective-space convention).

Generating sections define a morphism: if $s_0,\dots ,s_p$ are global sections generating an invertible sheaf $G$, there is a unique $k$-morphism $\phi :X\to {\mathbb{P}}_k^p$ with ${\phi }^{\ast }\mathcal{O}(1)\cong G$ and ${\phi }^{\ast }(x_i)=s_i$ (Generating line-bundle sections define a morphism to projective space).

Graphs and immersions: for a $k$-morphism $u:X\to Y$ with $Y$ separated over $k$ the graph ${\mathrm{\Gamma }}_u:X\to X{\times }_{k}Y$ is a closed immersion, because the defining square is a base change of the diagonal ${\mathrm{\Delta }}_{Y\mathrm{/}k}$ (The graph is a pullback of the diagonal, Separated morphism of schemes) and base changes of closed immersions are closed immersions (Closed immersions are affine quotients and survive base change); the projective space ${\mathbb{P}}_k^p$ is proper over $k$, hence separated (Finite-dimensional projective space is proper over every base, Separated morphism of schemes), and a composite of closed immersions is a closed immersion, since the composite of homeomorphisms onto closed subsets is again one and a composite of surjective sheaf maps is surjective (Closed immersions of schemes). The canonical swap $X{\times }_k{\mathbb{P}}_k^p\to {\mathbb{P}}_k^p{\times }_{k}X$ is an isomorphism, so it preserves closed immersions.

The Segre embedding: with $p,q\ge 0$ and $P={\mathbb{P}}_k^p{\times }_k{\mathbb{P}}_k^q$ with projections ${\mathrm{p}\mathrm{r}}_1,{\mathrm{p}\mathrm{r}}_2$, there is a closed immersion $\sigma :P\hookrightarrow {\mathbb{P}}_k^{(p+1)(q+1)-1}$ with ${\sigma }^{\ast }\mathcal{O}(1)\cong {\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$ (Segre embedding and its line bundle).

The Axiom of Choice is inherited from the Proj, ample-embedding and Segre suppliers of [F2], [F4] and [F5]; the finitely many generating sections chosen in step 3.1 are a finite family, and no infinite selection occurs.

Given: a field $k$, an integral smooth projective surface $X$ over $k$, a canonical divisor $K_X$ with ${\mathcal{O}}_X(K_X)\cong {\omega }_X$, and a nonzero effective Cartier divisor $C\subseteq X$.

$X$ is proper over $k$ and Noetherian; the intersection product on invertible sheaves and Cartier divisors is defined by the alternating sum of Intersection numbers of Cartier divisors on a smooth projective surface, is symmetric and $\mathbb{Z}$-bilinear, and depends only on the linear equivalence classes of its entries (The surface intersection product is symmetric and bilinear). For an effective Cartier divisor $C$ and any Cartier divisor $D$ one has $C\cdot D={\mathrm{deg}}_C({\mathcal{O}}_X(D){\mathrm{\mid }}_C)$ with ${\mathrm{deg}}_C(\mathcal{N})=\chi (C,\mathcal{N})-\chi (C,{\mathcal{O}}_C)$ (Intersection with a curve is the degree of the restriction, Degree of an invertible sheaf on a proper one-dimensional scheme).

Serre duality: for every invertible ${\mathcal{O}}_X$-module $E$ and every $q$ the cup-product pairing $H^q(X,E)\times H^{2-q}(X,E^{\vee }\otimes {\omega }_X)\to k$ is perfect and all groups are finite-dimensional (Serre duality for locally free sheaves on a smooth projective variety). Consequently $\chi (X,E)=\chi (X,E^{\vee }\otimes {\omega }_X)$ and $\chi (X,{\omega }_X)=\chi (X,{\mathcal{O}}_X)$; moreover ${\mathcal{O}}_X(K_X)\cong {\omega }_X$, so ${\mathcal{O}}_X(K_X+D)\cong {\omega }_X\otimes {\mathcal{O}}_X(D)$ and $({\omega }_X\otimes {\mathcal{O}}_X(D){)}^{\vee }\cong {\mathcal{O}}_X(-K_X-D)$ for every Cartier divisor $D$ (The canonical divisor of a smooth projective surface, Dual of a line bundle is its tensor inverse, Tensor product of sheaves of modules, Invertible sheaves).

Structure sequence and additivity: for the nonzero effective Cartier divisor $C$ there is a short exact sequence of coherent ${\mathcal{O}}_X$-modules $0\to {\mathcal{O}}_X(-C)\to {\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_C\to 0$ (Effective Cartier divisors give a short exact sequence); the Euler characteristic is additive on short exact sequences of coherent modules on the proper scheme $X$ (Euler characteristic is additive in short exact sequences); and $\chi (X,i_{\ast }{\mathcal{O}}_C)=\chi (C,{\mathcal{O}}_C)$ (Euler characteristic of a coherent sheaf, the projection-formula identification of Intersection with a curve is the degree of the restriction in the effective case). Hence $\chi (X,{\mathcal{O}}_X)-\chi (X,{\mathcal{O}}_X(-C))=\chi (C,{\mathcal{O}}_C)$.

The divisor $-K_X-C$ is a Cartier divisor and ${\mathcal{O}}_X(-K_X-C)\cong ({\omega }_X\otimes {\mathcal{O}}_X(C){)}^{\vee }$, while ${\mathcal{O}}_X(-C)\cong {\mathcal{O}}_X(C{)}^{\vee }$; the canonical divisor is a Cartier divisor and linear equivalence may be replaced by the associated invertible sheaves in all intersection computations (The canonical divisor of a smooth projective surface, Cartier divisor, Invertible sheaf of cartier divisor).

The Axiom of Choice is inherited from the Serre-duality and Euler-characteristic suppliers of [F2] and [F3]; the divisor $C$ and the canonical divisor are given data.

Given: a field $k$, an integral smooth projective surface $X$ over $k$, a canonical divisor $K_X$, an ample invertible sheaf $H$, and an invertible sheaf $M$ with $M\cdot H>K_X\cdot H$.

Serre duality: for the invertible sheaf $M$ the pairing $H^2(X,M)\times H^0(X,M^{\vee }\otimes {\omega }_X)\to H^2(X,{\omega }_X)\stackrel{t_X}{\to }k$ is perfect and both groups are finite-dimensional (Serre duality for locally free sheaves on a smooth projective variety); hence $h^2(X,M)=h^0(X,{\omega }_X\otimes M^{\vee })$ and $H^2(X,M)=0$ if and only if $H^0(X,{\omega }_X\otimes M^{\vee })=0$ (Euler characteristic of a coherent sheaf, Invertible sheaves, Tensor product of sheaves of modules).

Positivity of ample classes: if $D$ is a nonzero effective Cartier divisor, then $H\cdot D>0$; and if an invertible sheaf $\mathcal{N}$ has a nonzero global section whose zero scheme is empty, then $\mathcal{N}\cong {\mathcal{O}}_X$ (Ample divisors meet nonzero effective divisors positively).

Zero schemes of sections: a nonzero global section of an invertible sheaf on the integral scheme $X$ is regular, its zero scheme $Z(s)$ is an effective Cartier divisor with ${\mathcal{O}}_X(Z(s))\cong \mathcal{N}$, and $Z(s)$ is empty if and only if $s$ is nowhere vanishing (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Zero scheme of a line-bundle section).

The intersection numbers $K_X\cdot H$ and $M\cdot H$ are computed through the associated invertible sheaves ${\omega }_X$ and $M$; they depend only on the linear equivalence classes and are additive, so that $(K_X-M)\cdot H=K_X\cdot H-M\cdot H$ (The canonical divisor of a smooth projective surface, Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear).

The Axiom of Choice is inherited from the Serre-duality and ample-positivity suppliers of [F1] and [F2]; the section used below, if it exists, is a single object.

Given: a field $k$, an integral smooth projective surface $X$ over $k$, a canonical divisor $K_X$, and an invertible sheaf $\mathcal{L}$.

The intersection product is defined by the alternating sum ${\mathcal{L}}_1\cdot {\mathcal{L}}_2=\chi (X,{\mathcal{O}}_X)-\chi (X,{\mathcal{L}}_1^{\vee })-\chi (X,{\mathcal{L}}_2^{\vee })+\chi (X,{\mathcal{L}}_1^{\vee }\otimes {\mathcal{L}}_2^{\vee })$ of Intersection numbers of Cartier divisors on a smooth projective surface, is symmetric and $\mathbb{Z}$-bilinear, and satisfies ${\mathcal{L}}_1^{\vee }\otimes {\mathcal{L}}_1\cong {\mathcal{O}}_X$ (The surface intersection product is symmetric and bilinear, Dual of a line bundle is its tensor inverse, Tensor product of sheaves of modules, Invertible sheaves).

The canonical sheaf satisfies ${\mathcal{O}}_X(K_X)\cong {\omega }_X={\bigwedge }^2{\mathrm{\Omega }}_{X\mathrm{/}k}^1$, and $K_X\cdot \mathcal{L}={\omega }_X\cdot \mathcal{L}$ for every invertible $\mathcal{L}$ (The canonical divisor of a smooth projective surface).

Serre duality: for every invertible sheaf $E$ the cup-product pairing $H^q(X,E)\times H^{2-q}(X,E^{\vee }\otimes {\omega }_X)\to k$ is perfect with finite-dimensional groups, so $\chi (X,E)=\chi (X,E^{\vee }\otimes {\omega }_X)$; in particular $\chi (X,{\omega }_X)=\chi (X,{\mathcal{O}}_X)$ (Serre duality for locally free sheaves on a smooth projective variety, Euler characteristic of a coherent sheaf).

The Axiom of Choice is inherited from the Serre-duality and Euler-characteristic suppliers of [F3]; the sheaf $\mathcal{L}$ and the divisor $K_X$ are given data.