Hilbert schemes of 8 points

Abstract

The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which generically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and d >= 4. In the simplest case of reducibility, the component R^4_8 \subset H^4_8 is defined by a single explicit equation which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of H_n^d which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example.

28 pages; Rewrote introduction and reorganized parts of the paper, some minor errors have been fixed

Cited by in corpus (38)

• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• Irreducibility of the Gorenstein loci of Hilbert schemes via ray families
• Elementary components of Hilbert schemes
• Smooth and irreducible multigraded Hilbert schemes
• Fibers of Generic Projections
• Murphy's Law for Hilbert function strata in the Hilbert scheme of points
• A syzygetic approach to the smoothability of zero-dimensional schemes
• Detaching embedded points
• Macaulay-like marked bases
• Commuting matrices and the Hilbert scheme of points on affine spaces
• Deformations of zero-dimensional schemes and applications
• The Hilbert scheme of points and its link with border basis
• Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system
• The Hilbert schemes of points on surfaces with rational double point singularities
• On the tangent space to the Hilbert scheme of points in P3
• Schémas de Hilbert invariants et théorie classique des invariants
• Components of Gröbner strata in the Hilbert scheme of points
• Irreducibility of the Gorenstein locus of the punctual Hilbert scheme of degree 10
• Syzygies in Hilbert schemes of complete intersections
• The space of generically étale families
• The motive of the Hilbert scheme of points in all dimensions
• On the smoothness of lexicographic points on Hilbert schemes
• The singularities of the principal component of the Hilbert scheme of points
• Identifying limits of ideals of points in the case of projective space
• Border Waring Rank via Asymptotic Rank
• The Hilbert scheme of 11 points in A^3 is irreducible
• Finite schemes and secant varieties over arbitrary characteristic
• Local finite dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable
• On extension of the motivic cohomology beyond smooth schemes
• A Riemann--Kempf singularity theorem for higher rank Brill--Noether loci
• Combinatorial duality of Hilbert schemes of points in the affine plane
• VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A^6
• The Gröbner fan of the Hilbert scheme
• Bounds on complexity of matrix multiplication away from CW tensors
• Symmetric Ideals and Invariant Hilbert Schemes
• The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage
• Invariants of nested Hilbert and Quot schemes on surfaces
• On the Gorenstein locus of some punctual Hilbert schemes