An enriched degree of the Wronski

TL;DR

This paper extends the classical Wronski map degree to an enriched setting over all even dimensions using $ extbf{A}^1$-homotopy theory, providing new algebraic invariants for intersection problems.

Contribution

It introduces an enriched degree of the Wronski map valued in the Grothendieck-Witt ring for all even dimensions, utilizing $ extbf{A}^1$-homotopy theory, and analyzes local intersection contributions.

Findings

Enriched degree computed in all even dimensions.

Local contributions relate to determinantal Plücker coordinate relationships.

Application to classical intersection problems like lines meeting multiple lines.

Abstract

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 for all $m$ and $p$ even, valued in the Grothendieck-Witt ring of a field, using machinery from $\mathbf{A}^1$-homotopy theory. We further demonstrate in all parities that the local contribution of an $m$-plane is a determinantal relationship between certain Pl\"ucker coordinates of the $p$-planes it intersects.