A Borel open cover of the Hilbert scheme

TL;DR

This paper introduces a new, smaller open cover for the Hilbert scheme using marked schemes over Borel-fixed ideals, enabling more practical computations by reducing complexity and degree of defining equations.

Contribution

It presents a novel open cover of the Hilbert scheme based on marked schemes, significantly reducing the number of open subsets and the complexity of defining equations.

Findings

Open cover with fewer subsets based on Borel-fixed ideals

Equations defining the open subsets have degree ≤ d+2

Constructive proofs using a term order free polynomial reduction process

Abstract

Let $p(t)$ be an admissible Hilbert polynomial in $\PP^n$ of degree $d$. The Hilbert scheme $\hilb^n_p(t)$ can be realized as a closed subscheme of a suitable Grassmannian $ \mathbb G$, hence it could be globally defined by homogeneous equations in the Plucker coordinates of $ \mathbb G$ and covered by open subsets given by the non-vanishing of a Plucker coordinate, each embedded as a closed subscheme of the affine space $A^D$, $D=\dim(\mathbb G)$. However, the number $E$ of Plucker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of $\hilb^n_p(t)$, we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than $E$. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree $\leq d+2$ in their natural embedding in $\Af^D$. Furthermore we find new embeddings in affine spaces of far lower dimension than $D$, and characterize those that are still defined by equations of degree $\leq d+2$. The proofs are constructive and use a polynomial reduction process, similar to the one for Grobner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.