## Laurent Evain ; Mathias Lederer - Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors

epiga:5618 - Épijournal de Géométrie Algébrique, April 29, 2021, Volume 5 - https://doi.org/10.46298/epiga.2021.volume5.5618
Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors

Authors: Laurent Evain ; Mathias Lederer

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.

Volume: Volume 5
Published on: April 29, 2021
Submitted on: July 10, 2019
Keywords: Mathematics - Algebraic Geometry,14C05