|Cohomology and Frobenius
June 11th - July 1st, 2023
Algebraic Geometry and Commutative Algebra
Sponsored by UNICT, PRIN 2020 "Squarefree Gr÷bner degenerations, special varieties and related topics", GNSAGA, DMI UNICT
Luis Nu˝ez-Betancourt (CIMAT, Mexico)
Eamon Quinlan-Gallego (University of Utah, USA)
|Cohomology of line
bundles on flag varieties
Claudiu Raicu (Notre Dame University, USA)
Alessio Sammartano (Politecnico di Milano, Italy)
Keller VandeBogert (Notre Dame University, USA)
|Pragmatic Home Page
(Promotion of Research in Algebraic Geometry for MAThematicians in
Isolated Centres) is a project for stimulating researches in Algebraic
Geometry and Commutative Algebra among young people.
The Frobenius morphism has proven to be a powerful tool in commutative algebra. Peskine and Szpiro proved Auslander's zero divisor conjecture and answered Bass' question regarding Cohen-Macaulay rings using this map. They also provided a series of important results regarding the structure of local cohomology. Hochster and Roberts proved that rings of invariants are Cohen-Macaulay. Later, Hochster and Huneke developed tight closure theory and used it to solve several homological conjectures in prime characteristic. Since then, the use of the Frobenius map has grown to touch other areas such as differential algebra, combinatorics, representation theory and birational geometry. The first lectures will focus on the basics on the Frobenius maps, its splittings, and local cohomology. Then, we focus on the use of Frobenius in singularity theory and combinatorics. We will also discuss connections with D-modules. There will also be an introduction to Macaulay2 packages in prime characteristic, and related topics. The list of open problems to discuss will try to reflect the diversity of uses of the Frobenius map.
Cohomology of line bundles
The goal of these lectures is to discuss the problem of computing cohomology of line bundles on flag varieties, and to explore various applications to the study of homological invariants in commutative algebra and algebraic geometry. Over fields of characteristic zero, the cohomology calculation is well-understood, and is the subject of the celebrated Borel-Weil-Bott theorem. In positive characteristic however, the description of cohomology is largely unknown, even when it comes to deciding its vanishing and non-vanishing behavior. The question of computing cohomology turns out to be quite versatile – any reasonable restrictions, either on the line bundles considered, or on the flag varieties themselves, lead to special cases that are of interest on their own, and the participants will have the chance to examine a number of particular scenarios. The applications we envision include, but are not limited to: Castelnuovo-Mumford regularity for powers and symbolic powers of determinantal varieties, Hilbert functions of Koszul modules, or the homology of certain arithmetic Koszul-type complexes.