Quasi-projectivity of normal varieties

TL;DR

This paper proves that normal varieties have finitely many maximal quasi-projective open subvarieties and generalizes the Chevalley-Kleiman criterion, linking quasi-projectivity to the containment of finite sets in affine open subvarieties.

Contribution

It establishes the finiteness of maximal quasi-projective open subvarieties in normal varieties and extends the projectivity criterion to a broader class of varieties.

Findings

Normal varieties contain finitely many maximal quasi-projective open subvarieties.

A normal variety is quasi-projective iff every finite subset lies in an affine open.

The proof uses a strategy of Wlodarczyk and results from Boissiere, Gabber, and Serman.

Abstract

We prove that a normal variety contains finitely many maximal quasi-projective open subvarieties. As a corollary, we obtain the following generalization of the Chevalley-Kleiman projectivity criterion : a normal variety is quasi-projective if and only if every finite subset is contained in an affine open subvariety. The proof builds on a strategy of Wlodarczyk, using results of Boissiere, Gabber and Serman.