Fall 2026 @ Brown University - November 20, 2026

Constructing compactified Jacobians of curves and characterizing them is an old problem in algebraic geometry. In the case of nodal curves, which admit a dual graph, smoothable compactified Jacobians are completely characterized by V-stability conditions, which are assignments of integers to the biconnected subsets of the vertices of the dual graph. In this talk, we will explore the combinatorics of V-stability conditions. In particular, I will highlight a construction of stability conditions from an oriented matroid polytope, called a Lawrence polytope, associated with an orientation of the dual graph, and how this construction provides insight into the structure of all V-stability conditions and related objects such as degeneracy sets.

Positroid varieties are an important class of subvarieties of the Grassmannian which carry rich combinatorial structures from both Schubert geometry and matroid theory. Knutson-Lam-Speyer showed that they are normal, Cohen--Macaulay, and have rational singularities, and Billey-Weaver gave combinatorial criteria for smoothness. In this talk, we discuss their divisors and related geometric properties. We describe the Picard groups of positroid varieties and their generators, and give an explicit graph-theoretic criterion for when a positroid variety is Gorenstein. We show that being Gorenstein, Q-Gorenstein, and Fano are equivalent for positroid varieties. We further characterize when a Gorenstein positroid variety is arithmetically Gorenstein in the Plücker embedding. Joint with Ayah Almousa and Anders Buch.