Student Commutative Algebra Seminar

Let R be a singular ring, and D_R the ring of differential operators on R. Despite decades of study and many fruitful applications, D_R remains quite mysterious. In this talk, we will discuss connections between the algebraic description of differential operators on singular rings and the global geometry of Fano varieties. A recurring question in the literature is whether "nice" properties of differential operators (e.g., if R is a simple D_R-module, or "D-simple" for short) are implied by the "mildness" of the singularity. Specifically, we focus on the question of whether a Gorenstein ring with klt singularities must be D-simple. We answer this question in the negative, by giving several explicit examples, which arise as homogeneous coordinate rings of smooth Fano varieties X. The property of this ring being D-simple can be translated into the question of positivity (specifically, bigness) of the tangent bundle of X. I will discuss results on the bigness of the tangent bundle of del Pezzo surfaces and recent work of Höring, Liu, and Shao which completed the surface; perhaps more importantly, I will point out how much is left to discover about bigness of the tangent bundle in higher dimensions. https://umich.zoom.us/j/99835724541 Speaker(s): Devlin Mallory (University of Michigan)