On the $\mathrm{PGL}_2$-equivariant intersection theory of $\mathrm{Gr}(2,4)$

TL;DR

This paper computes the $ ext{PGL}_2$-equivariant Chow ring of the stable locus of $ ext{Gr}(2,4)$ and explores the quotient stack structure, highlighting challenges in extending to the full Grassmannian.

Contribution

It provides an explicit description of the $ ext{PGL}_2$-equivariant Chow ring of the stable locus of $ ext{Gr}(2,4)$ and relates the quotient stack to an open subset of $ ext{P}^1$ by $S_4$.

Findings

Determined the $ ext{PGL}_2$-equivariant Chow ring of $ ext{Gr}(2,4)^s$.

Presented the quotient stack as an open subset of $ ext{P}^1$ modulo $S_4$.

Identified difficulties in computing the full $ ext{PGL}_2$-equivariant Chow ring of $ ext{Gr}(2,4)$.

Abstract

We determine the $\mathrm{PGL}_2$-equivariant Chow ring of $\mathrm{Gr}(2,4)^s$, the $\mathrm{PGL}_2$-stable locus of $\mathrm{Gr}(2,4)$, over any algebraically closed based field of characteristic not equal to 2 or 3. In the process, we demonstrate that the quotient stack $[\mathrm{Gr}(2,4)^s/\mathrm{PGL}_2]$ can be presented as the quotient of an open subset of $\mathbb{P}^1$ by a suitably chosen $S_4\leq \mathrm{PGL}_2$. We also discuss some apparent difficulties with computing the full $\mathrm{PGL}_2$-equivariant Chow ring of $\mathrm{Gr}(2,4)$.