Immaculate line bundles on toric varieties

TL;DR

This paper studies immaculate line bundles on toric varieties, providing a geometric approach to identify all such bundles and exploring their structure and properties within the class group.

Contribution

It introduces a method to determine the locus of immaculate divisors on toric varieties using momentum polytopes and analyzes their structure in specific cases.

Findings

The locus of immaculate line bundles includes several linear strata.

A new notion of relative immaculacy is introduced, explaining some strata.

The method is applied to smooth toric varieties of Picard rank two and three.

Abstract

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.