Vanishing of top cohomology past the canonical threshold

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.