A stronger version of the Kleiman-Chevalley projectivity criterion (Q2792252)

Let \(X\) be a normal algebraic variety. It was conjectured by Chevalley (for complete varieties) and proved by \textit{O. Benoist} [Int. Math. Res. Not. 2013, No. 17, 3878--3885 (2013; Zbl 1312.14022)] that projectivity of \(X\) follows from the property that every finite set on \(X\) is contained in an open affine subset of \(X\). The number \(a(X)\) (defined as the supremum of \(n \in \mathbb{N}\) such that any \(n\)--points of \(X\) are contained in an affine open subset of \(X\)) has been of importance in the history of the proof of this conjecture (see the Introduction of the paper under review and references therein). In particular, there exist several results showing that if \(a(X)\) is big enough then \(X\) is projective. The main result of this paper improves them showing (see Thm. 1.2) that if \(X\) is complete and \(\mathbb{Q}\)-factorial and \(a(X)\) is bigger than or equal to the Picard number of \(X\) then \(X\) is projective. This bound is not known to be sharp but there exist smooth complete nonprojective algebraic varieties for which \(a(X)\) equals its Picard number minus two.