Geometry Seminar

Date: March 11, 2022

Time: 4:00PM - 4:50PM

Location: BLOC 302

Speaker: Thomas Brazelton, University of Pennsylvannia

Title: An enriched degree of the Wronski

Abstract: Given $mp$ different $m$-planes in general position in $(m+p)$-dimensional space, a classical problem is to ask how many $p$-planes intersect all of them. For example when $m=p=2$, this is precisely the question of "lines meeting four lines in 3-space" after projectivizing. The Shapiro conjecture asserts that all solutions to a real Schubert problem of this type will be real in the setting where the planes are chosen to osculate a rational normal curve. In this setting, the Brouwer degree of the Wronski map provides an answer to this question, first computed by Schubert over the complex numbers and Eremenko and Gabrielov over the reals. We provide an enriched degree of the Wronski for all $m$ and $p$ even, valued in the Grothendieck--Witt ring of an arbitrary field. We further demonstrate in all parities that the local contribution of a $p$-plane is a determinantal relationship between certain Plücker coordinates of the $m$-planes it intersects.