Incidence relations among the Schubert cells of equivariant Hilbert Schemes

TL;DR

This paper studies the stratification of equivariant Hilbert schemes of points on the plane, providing a necessary condition for the incidence relations among Schubert cells, extending known results from Grassmannians.

Contribution

It introduces a necessary condition for the closure relations among Schubert cells in equivariant Hilbert schemes, generalizing the classical Grassmannian case.

Findings

Provides a necessary condition for cell closure relations.

Extends incidence relation understanding from Grassmannians to equivariant Hilbert schemes.

Highlights the complexity of cell interactions in the equivariant setting.

Abstract

Let HH_{ab}(H) be the equivariant Hilbert scheme parametrizing the zero dimensional subschemes of the affine plane k^2, fixed under the one dimensional torus T_{ab}={(t^{-b},t^a), t\in k^*} and whose Hilbert function is H. This Hilbert scheme admits a natural stratification in Schubert cells which extends the notion of Schubert cells on Grassmannians. However, the incidence relations between the cells become more complicated than in the case of Grassmannians. In this paper, we give a necessary condition for the closure of a cell to meet another cell. In the particular case of Grassmannians, it coincides with the well known necessary and sufficient incidence condition. There is no known example showing that the condition wouldn't be sufficient.