Jarek Buczyński: Immaculate line bundles on toric varieties

Abstract: A sheaf on an algebraic variety is called 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 talk we will focus on line bundles on smooth projective toric varieties. First, we present a possibility of understanding their cohomology groups in terms of their momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This can be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. Moreover, the locus of immaculate line bundles contains several linear strata of varying dimensions. A notion of relative immaculacy with respect to certain contraction morphisms is stronger than plain immaculacy and provides an explanation of some of these linear strata. The talk is based on a joint work with Klaus Altmann, Lars Kastner, and Anna-Lena Winz, see also arxiv.org/abs/1808.09312 .