Given mp different p-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 Brouwer degree of the Wronski map provides an answer to this general question, first computed by Schubert over the complex numbers and Eremenko and Gabrielov over the reals. We provide an enriched degree of the Wronski map for all m and p even, valued in the Grothendieck--Witt ring of a field, using machinery from A¹-homotopy theory. We further demonstrate in all parities that the local contribution of an m-plane is a determinantal relationship between certain Plucker coordinates of the p-planes it intersects.

This paper was written with support from an NSF Graduate Research Fellowship (DGE-1845298) and NSF Postdoctoral Fellowship (DMS-2303242).