On non-projective complete toric varieties

TL;DR

This paper demonstrates that certain smooth non-projective complete toric threefolds with low Picard number can be transformed into projective varieties through a finite sequence of flops or anti-flips, revealing their underlying projective structure.

Contribution

It shows that all smooth non-projective complete toric threefolds with Picard number at most five become projective after specific birational transformations.

Findings

Smooth non-projective toric threefolds with Picard number ≤ 5 become projective after flops or anti-flips.

Every complete toric variety is isomorphic in codimension one to a projective toric variety.

The process of flops or anti-flips can convert certain non-projective varieties into projective ones.

Abstract

For every complete toric variety, there exists a projective toric variety which is isomorphic to it in codimension one. In this paper, we show that every smooth non-projective complete toric threefold of Picard number at most five becomes projective after a finite succession of flops or anti-flips.