Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors

TL;DR

This paper extends the concept of Bialynicki-Birula strata to higher-dimensional Hilbert schemes, introducing a schematic structure called the Bialynicki-Birula scheme, and proves its properties and relations to monic functors.

Contribution

It introduces the Bialynicki-Birula scheme as a natural schematic structure in higher dimensions and proves its representability and inclusion properties.

Findings

Bialynicki-Birula scheme is schematically included in certain fibers under torus actions.

Monic functors parametrizing ideals with prescribed initial ideals are representable.

The scheme generalizes known smooth strata in dimension 2 to higher dimensions.

Abstract

The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ are smooth in dimension $d=2$. We prove that there is a schematic structure in higher dimensions, the Bialynicki-Birula scheme, which is natural in the sense that it represents a functor. Let $\rho_i:H^n(\mathbb{A}^d)\rightarrow {\rm Sym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$ coordinate. We prove that a Bialynicki-Birula scheme associated with an action of a torus $T$ is schematically included in the fiber $\rho_i^{-1}(0)$ if the ${i}^{th}$ weight of $T$ is non-positive. We prove that the monic functors parametrizing families of ideals with a prescribed initial ideal are representable.