QM Research Seminar: Fundamental groups of complements of wobbly divisors in intersections of two quadrics

Speaker: Yuki Matsubara (University of Southern Denmark)

Abstract: 
A smooth complete intersection X of two quadrics in P^{n+2}  admits an interpretation as a moduli space of bundles.For n = 2, it parametrizes stable rank 2 parabolic bundles on P^1 with five marked points, while for n = 3 it parametrizes stable rank 2 bundles of odd degree with fixed determinant on a genus 2 curve. For general n, it can be interpreted as a moduli space of semistable twisted Spin-bundles. 

In my talk, I will discuss the topology of the very stable locus X \ W, where W is the wobbly locus, consisting of bundles that admit a nonzero nilpotent Higgs field. The importance of the very stable locus in the geometric Langlands correspondence was emphasized by Laumon. 

I will explain that W is an irreducible divisor for n >2 and how to compute H_1(X \ W, Z).Using a root stack description, we identify the fundamental group $\pi_1(X \ W)$ with the kernel of a monodromy homomorphism from an orbifold mixed braid group.The Reidemeister—Schreier method then yields an explicit presentation of this fundamental group in every dimension. 

​​​​​​​This talk is based on joint work with S. Bhattacharya (https://arxiv.org/abs/2609.31421).