A Lefschetz hyperplane theorem for Mori dream spaces

TL;DR

This paper proves a Lefschetz-type theorem for Mori dream spaces, showing that certain ample divisors inherit the Mori dream space property and related geometric structures, enabling new constructions of such spaces.

Contribution

It establishes conditions under which smooth ample divisors of Mori dream spaces are also Mori dream spaces, extending the understanding of their geometric and combinatorial properties.

Findings

Smooth ample divisors of Mori dream spaces are also Mori dream spaces under certain GIT conditions.

The GIT condition is stable under products and projective bundles of line bundles.

In toric cases, the GIT condition corresponds to the fan being 2-neighborly.

Abstract

Let X be a smooth Mori dream space of dimension at least 4. We show that, if X satisfies a suitable GIT condition which we call "small unstable locus", then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Neron-Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.