Ample divisors meet nonzero effective divisors positively

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.