Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 4

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

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n4.a12


Klaus Altmann (Institut für Mathematik, Freie Universität, Berlin, Germany)

Jarosław Buczyński (Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland; and Faculty of Mathematics, Computer Science and Mechanicsm University of Warsaw, Poland)

Lars Kastner (Institut für Mathematik, Technische Universität Berlin, Germany)

Anna-Lena Winz (Institut für Mathematik, Freie Universität, Berlin, Germany)


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