The Locus of Points of the Hilbert Scheme with Bounded Regularity

Abstract

In this paper we consider the Hilbert scheme parameterizing subschemes of ℙ ⁿ with Hilbert polynomial p(t), 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 and, for every s ≥ r′, we describe it as a locally closed subscheme of the Grasmannian given by a set of equations of degree ≤deg(p(t)) +2 and linear inequalities in the coordinates of the Plücker embedding.

Key Words:

• Borel-fixed ideal
• Castelnuovo–Mumford regularity
• Hilbert scheme

2010 Mathematics Subject Classification:

• 14C05
• 14Q20

ACKNOWLEDGMENTS

The authors thank the referee for several very important suggestions that improved the quality of the paper.

Notes

Communicated by S. Kleiman.