1111.2007Nov 8, 2011v2: new proofs relying on the functorial definition of the Hilbert scheme. v3: Sections reorganized, new self-contained proof of the representability of the Hilbert functor with bounded regularity (Section 6)

The locus of points of the Hilbert scheme with bounded regularity

Edoardo Ballico, Cristina Bertone, Margherita Roggero

Algebraic Geometry

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Abstract

In this paper we consider the Hilbert scheme $Hilb_{p(t)}^n$ parameterizing subschemes of $P^n$ with Hilbert polynomial $p(t)$

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, and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer

r'

. This locus is an open subscheme of

Hilb_{p(t)}^n

and, for every

s\geq r'

, we describe it as a locally closed subscheme of the Grasmannian

Gr_{p(s)}^{N(s)}

given by a set of equations of degree

\leq \mathrm{deg}(p(t))+2

and linear inequalities in the coordinates of the Plücker embedding.

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Nearby in the stack

The locus of points of the Hilbert scheme with bounded regularity

Claims

Methods

Limitations

Nearby in the stack

The singularities of the principal component of the Hilbert scheme of points

Unexpected but recurrent phenomena for Quot and Hilbert schemes of points

The Hilbert scheme of a pair of linear spaces is a Mori dream space

New elementary components of the Gorenstein locus of the Hilbert scheme of points

The Calysto Scheme Project

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The Calysto Scheme Project

Compressive Domains and a Bound for the Number of Components of the Fixed Locus of a Self-Map of the Berkovich Line