The equivariant Chow rings of quot schemes

Abstract

We give a presentation for the (integral) torus-equivariant Chow ring of the quot scheme, a smooth compactification of the space of rational curves of degree d in the Grassmannian. For this presentation, we refine Evain's extension of the method of Goresky, Kottwitz, and MacPherson to express the torus-equivariant Chow ring in terms of the torus-fixed points and explicit relations coming from the geometry of families of torus-invariant curves. As part of this calculation, we give a complete description of the torusinvariant curves on the quot scheme and show that each family is a product of projective spaces.

A positive Monk formula in the S 1 -equivariant cohomology of type A Peterson varieties

“…The original theory builds combinatorial tools to compute the T -equivariant cohomology ring of a T -space X that satisfies certain technical conditions. This influential theory and its many consequences have been extensively generalized and used since [4,11,13,[15][16][17]19,20,22]. In particular, extensions of GKM theory apply to many of the generalized equivariant cohomology theories mentioned above.…”

Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity

“…Bifet's formula is later generalized to arbitrary vector bundles [6] (in place of the trivial vector bundle O ⊕d X ), and the nested Quot schemes [48]. Other works along this line consider high-rank-quotient versions of the Quot scheme [13,15,56] (where (equivariant) Chow rings are often computed as well) or the case where X is a smooth surface [39,52].…”

The T-Graph of a Multigraded Hilbert Scheme

“…The standard set-up of [GKM98] to compute cohomology from the T -graph of a variety requires that the one-dimensional orbits be isolated, which need not be the case for multigraded Hilbert schemes. However one could still hope to deduce information about the cohomology in these cases; see for example [BCS08,Eva07].…”