Hilbert Schemes of 8 Points in A

The Hilbert scheme H n of n points in A d contains an irreducible component Rdn which generically represents n distinct points in A. We show that when n is at most 8, the Hilbert scheme H n is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 8 ⊂ H 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 H n we study the closed subschemes which correspond to homogeneous ideals with a fixed Hilbert function. We describe the components of these subschemes when the colength is at most 8. In particular, we find the minimal such example which is reducible. The Hilbert scheme H n of n points in A d parametrizes 0-dimensional, degree n subschemes of A. Equivalently, the k-valued points of H n parametrize ideals I ⊂ S = k[x1, . . . , xd] such that S/I is an n-dimensional vector space over k. We will use I to denote both an ideal in S and a point in H n. Since its introduction, the Hilbert scheme of points has been an active area of research and is the natural context for many fundamental questions about 0-dimensional subschemes of affine space. The question motivating this paper is how to characterize the 0-dimensional subschemes which are limits of distinct points. This question can be stated in terms of the geometry of the Hilbert scheme. Ideals of distinct points form a dn-dimensional open subset of H n. Its closure, R d n, is a component of the Hilbert scheme. This component parametrizes all ideals which are limits of distinct points. Ideals belonging to R n are called smoothable, and we will refer to R n as the smoothable component. It is well-known that the smoothable component is the only component of the Hilbert scheme of points in the plane [Fog68]. Perhaps surprisingly, this result is no longer true when d is at least 3. For any d at least 3 and n sufficiently large the Hilbert scheme of points is always reducible [Iar72]. Note that if H n is reducible, then so is H n for all n ′ > n. Thus for fixed d, it is interesting to ask for the minimal n such that the Hilbert scheme is reducible. Hilbert schemes of small numbers of points have been previously studied and [Iar87, p. 311] provides an overview of results. To prove the irreducibility of Hilbert schemes of small numbers of points, one could classify all isomorphism types of algebras with vector space dimension at most n and then construct degenerations. Mazzola used this approach, and his results imply the irreducibility of H n for n at most 7 [Maz80]. Taking a fundamentally different approach from Mazzola, we determine the components of H n for n at most 8 over any field of characteristic not 2 or 3. Our starting point is the observation that Artin local algebras naturally decompose into families over the schemes parametrizing homogeneous ideals with a fixed Hilbert function. This decomposition of local algebras is coarser than by isomorphism type but allows us to use geometric tools to extend Mazzola’s smoothability results. The schemes parametrizing homogeneous ideals with a fixed Hilbert function ~h = (h0, h1, . . . ) are the special case of multigraded Hilbert schemes obtained by taking the standard grading on S [HS04]. We will refer to these as standard graded Hilbert schemes and denote them H ~h. We describe their components when ∑ hi is at most 8. In particular, we show that H 3 (1,3,2,1) is the minimal standard graded Hilbert scheme which is reducible. A tangent space computation shows that the Hilbert scheme H 8 has at least two irreducible components [EI78]. In this paper we show that the intersection between these two components is irreducible and has an elegant description as the vanishing locus of a certain Pfaffian. This equation gives a simple polynomial condition for determining whether or not a given ideal is smoothable. DAC was supported by an NSF EMSW21 fellowship. DE was supported by an NDSEG fellowship. BV was supported by a Mentored Research Award. 1 1. Statement of Results Unless otherwise specified, k will denote a field of characteristic not 2 or 3. Theorem 1.1. Suppose n is at most 8 and d is any positive integer. Then the Hilbert scheme H n is reducible if and only if n = 8 and d ≥ 4, in which case it consists of exactly two irreducible components: (1) the smoothable component, of dimension 8d (2) a component of dimension 8d− 7, which we denote G8 Both of these components are rational. In the case of d = 4, n = 8, we can describe the situation more explicitly. Let S = k[x, y, z, w] and let S 2 be the space of symmetric bilinear forms on the vector space S1. Then, the component G 4 8 is isomorphic to A ×Gr(3, S 2 ), where Gr(3, S ∗ 2) denotes the Grassmannian of 3-dimensional subspaces of S ∗ 2 . Theorem 1.2. The intersection R 8 ∩ G 4 8 is a prime divisor on G 4 8. We have the following equivalent descriptions of R 8 ∩G 4 8: (1) Set Theoretic Since the intersection is integral, it is sufficient to describe R 8 ∩ G 4 8 as a subset of G8. For a point I ∈ G 4 8 ∼= A ×Gr(3, S 2 ) ∼= A ×Gr(7, S2) let V be the corresponding 7-dimensional subspace of S2. Then I ∈ G 4 8 belongs to the intersection if and only if the following skew-symmetric bilinear form 〈, 〉I is degenerate: 〈, 〉I : (S1 ⊗ S2/V ) ⊗2 → 3