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

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 ${\rho }_i:H^n({\mathbb{A}}^d)\to {\mathrm{Sym}}^n({\mathbb{A}}^1)$ 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.

Final version

References in corpus (1)

• On algebraic spaces with an action of G_m

Cited by in corpus (3)

• Computable Białynicki-Birula decomposition of the Hilbert scheme
• Combinatorial duality of Hilbert schemes of points in the affine plane
• On the cup product for Hilbert schemes of points in the plane