$PGL_2$-equivariant strata of point configurations in $\mathbb{P}^1$

Abstract

We compute the integral Chow ring of the quotient stack [(P 1 ) n /P GL 2 ], which contains M 0,n as a dense open, and determine a natural Z-basis for the Chow ring in terms of certain ordered incidence strata. We further show that all Z-linear relations between the classes of ordered incidence strata arise from an analogue of the WDVV relations in A • (M 0,n ).Next we compute the classes of unordered incidence strata in the integral Chow ring of the quotient stack [Sym n P 1 /P GL 2 ] and classify all 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.

Equivariant Degenerations of Plane Curve Orbits

“…We will let u, v be the chern roots of c 1 and c 2 . In the case X ⊂ P 1 is supported on at most three points, these are strata of coincident root loci, which were first computed in [FNR06] and generalized to P GL 2 -equivariant cohomology in [ST18]. Therefore, we only have to deal with the case where X is supported on at least four points, and we give two separate proofs.…”