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

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 $S_4\le {\mathrm{PGL}}_2$. We also discuss some apparent difficulties with computing the full ${\mathrm{PGL}}_2$-equivariant Chow ring of $\mathrm{Gr}(2,4)$.

34 pages. Comments welcome!