Pure and Applied Mathematics Quarterly
Volume 16 (2020)
Special Issue: In Honor of Prof. Gert-Martin Greuel’s 75th Birthday
Guest Editors: Igor Burban, Stanislaw Janeczko, Gerhard Pfister, Stephen S.T. Yau, and Huaiqing Zuo
Immaculate line bundles on toric varieties
Pages: 1147 – 1217
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 (generalised) 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 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.
toric variety, immaculate line bundle, splitting fan, toric varieties of Picard rank $3$, primitive collections
2010 Mathematics Subject Classification
Primary 14M25. Secondary 14F05, 14F17, 52B20.
Received 11 April 2019
Accepted 2 September 2019
Published 13 November 2020