Back to Commutative Algebra Seminar: Spring 2024
Commutative Algebra Seminar
Date: Friday, Apr 26, 2024
Time: 2:00PM - 3:00PM
Location: LCB 222
Unusual room
Devlin Mallory
University of Utah
Title |
Finite F-representation type for homogeneous coordinate rings |
Abstract |
Finite F-representation type is an important notion in characteristic-p commutative algebra, and is closely connected to the behavior of differential operators. Despite this, explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings (essentially, those of Calabi-Yau or general-type varieties) will fail to have finite F-representation type, via an analysis of their rings of differential operators. This illustrates a connection between the commutative-algebraic property of FFRT, and the algebro-geometric properties of positivity/negativity of tangent sheaves of varieties. This also provides instructive examples of the structure of the ring of differential operators for non-F-pure rings, which to this point have largely been unexplored. We will also discuss the case of Fano varieties: Recent work has provided non-toric smooth Fano varieties that do have FFRT (Grassmannians Gr(2,n) and the quintic del Pezzo surface). However, it seems unlikely that this will be true for all Fano varieties; we will present conjectural evidence that “in general” smooth Fano varieties will often fail to have FFRT. |
Commutative Algebra