Absolute ampleness by affine section opens

Let $X$ be a scheme and let $L$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves). Recall that $X$ is quasi-compact when its underlying topological space is quasi-compact (Quasi-compact and quasi-separated schemes).

Nonvanishing loci. For a global section $s\in \mathrm{\Gamma }(X,L^n)$ of a positive power with $n\ge 1$, the nonvanishing locus of $s$ is $X_s=\{x\in X:\text{the image of }s\text{ in the fibre }{L}^n{\otimes }_{{\mathcal{O}}_X}\kappa (x)\text{ is nonzero}\}\mathrm{.}$ This set is open: trivialise $L^n$ near a point, writing $s=fe$ with $e$ a frame. In the local ring at $x$, the fibre value is nonzero exactly when $f_x$ is a unit. A germ inverse of $f_x$ is represented by a section $g$ on a neighbourhood of $x$; since $(fg{)}_x=1$, the equality $fg=1$ holds after shrinking that neighbourhood. Thus $f_y$ is a unit throughout that neighbourhood, proving openness. The set $X_s$ does not depend on the chosen trivialisation: the vanishing of the fibre component is an intrinsic condition on the section.

Definition. An invertible ${\mathcal{O}}_X$-module $L$ is ample when the following two conditions hold.

1. $X$ is quasi-compact.
2. For every point $x\in X$ there exist an integer $n\ge 1$ and a global section $s\in \mathrm{\Gamma }(X,L^n)$ with $x\in X_s\text{and}{X}_s\text{ is an affine scheme.}$

The empty scheme is allowed: if $X=\mathrm{\varnothing }$ then $X$ is quasi-compact and condition (2) is vacuous, so every invertible sheaf on $X$ — there is exactly one, the zero module sheaf, which is invertible because the empty scheme has no points — is ample on $X$.

• Only positive powers occur. Condition (2) uses a section of $L^n$ with $n\ge 1$; a section of $L^0={\mathcal{O}}_X$ alone is never used, so the definition is stable under the conventions recorded at Twisting sheaf on Proj.
• The affine loci are a basis. If $L$ is ample, then for every point $x$ a nonvanishing affine locus contains $x$; since $X$ is quasi-compact, finitely many such affine loci cover $X$. Both facts are used later but neither is part of the definition: the definition asks for one locus per point.
• Equivalent formulations are results, not conventions. The global-generation characterisation of ampleness (Serre's criterion) is proved in Serre global-generation criterion for ampleness under the Axiom of Choice (The Axiom of Choice) and the Noetherian hypotheses stated there. Relative ampleness over a base is a separate notion, defined at Relative ampleness over an arbitrary base.