Abstracts Fall 2023

Speaker:  Charles Vial

Title:  On proper splinters in positive characteristic

Abstract: A commutative ring is called a splinter if any finite-module ring extension splits. By the direct summand conjecture, now a theorem due to André, every regular ring is a splinter. The notion of splinter can naturally be extended to schemes. In that context, every normal scheme in characteristic zero is a splinter. In contrast, Bhatt observed in his thesis that the splinter property for proper schemes in positive characteristic imposes strong constraints on the global geometry; for instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology. I will report on joint work with Johannes Krah where we describe further restrictions on the global geometry of proper splinters in positive characteristic. 

Speaker:  Benjamin Sung

Title: Stability conditions on non-commutative curves

Abstract: Recent progress in the theory of Bridgeland stability conditions has primarily concerned new constructions in higher dimensions, as well as applications to studying the geometry of moduli spaces of stable objects. On the other hand, the classification of triangulated categories with stability conditions and the study of their stability manifold is somewhat less understood. In this talk, I will describe the classification of categories in suitably low dimensions via their moduli space of Bridgeland-stable objects. By invoking a reconstruction result, I will demonstrate that such categories coincide with the bounded derived category of coherent sheaves on smooth projective curves. Finally, I will discuss the behavior and existence of Serre-invariant stability conditions on low-dimensional triangulated categories.  

Speaker:  Giovanni Inchiostro

Title:  Smooth weighted blowdowns

Abstract:  An analogue of blow-ups are weighted blow-ups. Those are transformations, in nature similar to a blow-up, but which are a bit more flexible. For example, weighted blow-ups give better algorithms for resolving singularities of algebraic varieties, and often appear in moduli spaces of interest. The price that one has to pay for the extra flexibility is that the result of a weighted blow-up might no longer be a variety, but rather an algebraic stack. Therefore one natural question is: when is an algebraic stack a weighted blow-up of a simpler space? My coauthors and I give some criteria for when this question has a positive answer. This is a joint work with Arena,  Di Lorenzo, Mathur, Obinna and Pernice. 

Speaker:  Tariq Syed

Title: Motivic cohomology of cyclic coverings

Abstract: Many examples of topologically contractible smooth affine complex varieties are given by cyclic coverings. In this talk, we explain some new results on the motivic cohomology of such cyclic coverings.