Quivers and equations a la Plücker for the Hilbert scheme

Abstract

Several moduli spaces parametrizing linear subspaces of the projective space are cut out by linear and quadratic equations in their natural embedding: Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper is to prove that a similar statement holds when one replaces linear subspaces with algebraic subschemes of the projective space. We exhibit equations of degree 1 and 2 that define schematically the Hilbert schemes ${\mathbf{Hilb}}_{{\mathbb{P}}^n}^p$ for all (possibly nonconstant) Hilbert polynomials $p$. The equations are reminiscent of the Plücker relations on the Grassmannians: they are built formally with permutations on indexes on the Plücker coordinates. Our method relies on a new construction of the Hilbert scheme as a quotient of a scheme of quiver representations.

Simplified and enhanced version. We prove that the bound $R>=r$ for the validity of our equations is sharp using considerations on Castelnuvo-Mumford-Gotzmann regularity. We explain the meaning of these equations when $R<r$. Generic points are not used any more, leading to the simplification of several involved technical details (see remark6.6 )

References in corpus (2)

• The Hilbert scheme of points and its link with border basis
• Hilbert schemes of points of ${\mathcal{O}}_{{\mathbb{P}}^1}(-n)$ as quiver varieties