Conventions and hypothesis bookkeeping for surface Riemann-Roch and Hodge index

Assume the Axiom of Choice where the cited cohomology, Euler-characteristic and duality suppliers do (The Axiom of Choice). The following conventions govern the items of this page.

Base field and smoothness. The theorems Riemann-Roch for smooth projective surfaces, The Hodge index theorem for smooth projective surfaces and Negative definiteness of the primitive part of the Neron-Severi space are stated for an integral smooth projective surface over an arbitrary field $k$, following Vakil's Theorem 20.2.13 and Exercise 20.2.B. Smoothness over $k$ (Smoothness over a field by geometric regularity) is used through the dualizing line bundle ${\omega }_X={\bigwedge }^2{\mathrm{\Omega }}_{X\mathrm{/}k}^1$ and Serre duality (Dualizing line bundle and trace datum of a smooth projective variety, Serre duality for locally free sheaves on a smooth projective variety); over an imperfect field it is strictly stronger than regularity, and no claim is made here for regular non-smooth surfaces. A smooth surface over a field is regular, so all intersection-theoretic items of Intersection numbers of Cartier divisors on a smooth projective surface apply.

Numerical versus linear equivalence. Numerical equivalence is defined by vanishing of all intersection numbers (Numerical equivalence and the Neron-Severi space of a surface); it is coarser than linear equivalence, and the equality case in the Hodge index theorem is numerical triviality, not linear triviality. The real Neron-Severi space ${\mathrm{N}}_{\mathbb{R}}^1(X)$ is finite-dimensional by the Neron-Severi theorem, which is not proved or used on this page. Accordingly, the signature statement in Negative definiteness of the primitive part of the Neron-Severi space is conditional on finiteness of $\rho (X)$ while its negative-definiteness statement is unconditional.

Positive-square hypothesis, and tools not used. The Hodge index theorem assumes $H\cdot H>0$, not ampleness of $H$; this generality is used in the companion example on the blowup, where the class $H={\pi }^{\ast }\mathcal{O}(1)$ of the blowup $\pi :X^{\mathrm{\prime }}\to {\mathbb{P}}^2$ of a rational point has $H\cdot H=1>0$ yet is not ample, since $H\cdot E=0$ for the exceptional curve $E$. Positivity of ample divisors against nonzero effective divisors is supplied by Ample divisors meet nonzero effective divisors positively and rests on the Hilbert-polynomial leading coefficient, not on any resolution or Bertini input. No Bertini theorem, no resolution of singularities and no Nakai-Moishezon criterion is invoked anywhere on this page: surface Riemann-Roch and both Hodge index statements are derived here from the cited duality, intersection and cohomology suppliers alone.