UBC Algebraic Geometry Seminar

We will meet to discuss the schedule for the semester.

Given a singularity with a crepant resolution, a symmetry of the derived category of coherent sheaves on the resolution may often be constructed using the formalism of spherical functors. I will introduce this, and discuss work in progress on general constructions of such symmetries for hypersurface singularities. This builds on previous results with Segal, and is inspired by work of Bodzenta-Bondal.

Degree 0 Donaldson-Thomas (DT) invariants of certain orbifolds correspond to counts of colored plane partitions, which are configurations of boxes in 3D space, where the boxes are colored in a way determined by the orbifold structure. We will explain this correspondence and compute equivariant K-theoretically refined degree 0 orbifold DT invariants for some orbifolds. The computation combines a technique called factorization with more classical combinatorial arguments.

We study the question of when a generic stably free module splits off a free summand of a given rank. This question has the following geometric interpretation due to M. Raynaud. Let V(r,n) denote the Stiefel variety GL(n)/GL(n-r) over a field k. There is a projection map V(r,n) -> V(1,n) given by “forgetting frames.” Raynaud showed that V(r,n) -> V(1,n) has a section if and only if the following holds: if P is any module over any k-algebra R with the property that P+R is isomorphic to R^n, then P has a free factor of rank r-1. Using machinery from A1-homotopy theory, we characterize those n for which the map V(r,n) -> V(1,n) has a section in the cases r <= 4 and under some restrictions on the base field. This is joint work with Ben Williams.

Let E be an elliptic curve over a number field F. We will discuss the structure of certain modules that arise in the Iwasawa theory of elliptic curves and their applications to the growth of Mordell-Weil ranks along infinite Galois extensions of F with noncommutative Galois groups.

A Kazhdan-Lusztig variety is the intersection of a Schubert variety with an affine cell in a flag manifold. Therefore, one can obtain local equations for Schubert varieties by using coordinates on the affine cell. Building on the work of Fulton and Knutson/Miller, in finite type A, Woo and Yong gave a Grobner basis for Kazhdan-Lusztig ideals. We will describe a generalization of their result to the affine type A flag manifold. We will define linear charts on affine flag manifolds using Bott-Samelson varieties, describe an analogue of Fulton's essential set, then use a result of Knutson on geometric vertex decompositions to show that our equations give a Grobner basis. This is joint work with Daoji Huang.

The Hecke orbit conjecture predicts that Hecke symmetries characterize the central foliation on Shimura varieties over an algebraically closed field $k$ of characteristic $p$. The original conjecture predicts that on the $\bmod p$ reduction of a Shimura variety, any prime-to- $p$ Hecke orbit is dense in the central leaf containing it, and was recently proved by a series of nice papers. However, the behavior of Hecke correspondences induced by isogenies between abelian varieties in characteristic $p$ and $p$-adically is significantly different from the behavior in characteristic zero and under the topology induced by Archimedean valuations. In this talk, we will formulate a p-adic analog of the Hecke orbit conjecture and investigate the $p$-adic monodromy of $p$-adic Galois representations attached to points of Shimura varieties of Hodge type. We prove a density theorem for the locus of formal neighborhood associated to the mod $p$ points of the Shimura variety whose monodromy is large and use it to deduce the non-where density of Hecke orbits under certain circumstances.

Matroids are combinatorial objects that abstract the notion of independence. The motivating examples are matroids arising from vector configurations. An important tool for studying matroids is the associated Chow ring. A recent result of Adiprasito, Huh, and Katz is that Chow rings arising from matroids behave "nicely" in ways similar to cohomology rings of smooth projective varieties. This result has geometric inspiration, a combinatorial proof, and powerful consequences in matroid theory. In this talk, we will introduce this geometrically inspired approach to matroid theory. Then, we will present a new computation for the "mixed volume of a matroid", that is, the matroid analog to the integration map on smooth projective varieties. This is joint work with Andy Hsiao, and Kalle Karu.

Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour beyond the smooth case remains less well understood. Building upon recent advancements by Bachmann, Elmanto, Morrow, and Bouis, we establish its A^1-homotopty invariance for a broader class of "smooth" schemes. This is part of ongoing work in collaboration with Tess Bouis.

Given a finite separable field extension and a linear algebraic group defined over the base field, we can study the "Norm Principle", which examines how the base change of the group behaves with respect to the norm map of the field extension. It remains an open question whether the norm principle holds for all linear algebraic groups. In this talk, we will recall a Galois cohomology approach to this problem, and discuss the norm principle for groups of type D_n, in particular over complete discretely valued fields.