$PG{L}_2$-equivariant strata of point configurations in ${\mathbb{P}}^1$

Abstract

We compute the integral Chow ring of the quotient stack $[({\mathbb{P}}^1{)}^n/PG{L}_2]$, which contains ${\mathcal{M}}_{0,n}$ as a dense open, and determine a natural $\mathbb{Z}$-basis for the Chow ring in terms of certain ordered incidence strata. We further show that all $\mathbb{Z}$-linear relations between the classes of ordered incidence strata arise from an analogue of the WDVV relations in $A^{\bullet }({\overline{\mathcal{M}}}_{0,n})$. Next we compute the classes of unordered incidence strata in the integral Chow ring of the quotient stack $[{\mathrm{Sym}}^n{\mathbb{P}}^1/PG{L}_2]$ and classify all $\mathbb{Z}$-linear relations between the strata via these analogues of WDVV relations. Finally, we compute the rational Chow rings of the complement of a union of unordered incidence strata.

39 pages, comments welcome! Added: upgraded computation of rational classes of strata of unordered points to integral classes