Hilbert scheme

In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes. Proj ( k [ x 0 , x 1 ] [ α , β , γ ] / ( α x 0 2 + β x 0 x 1 + γ x 1 2 ) ) ⊆ P x 0 , x 1 1 × P α , β , γ 2 {displaystyle { ext{Proj}}(k/(alpha x_{0}^{2}+eta x_{0}x_{1}+gamma x_{1}^{2}))subseteq mathbb {P} _{x_{0},x_{1}}^{1} imes mathbb {P} _{alpha ,eta ,gamma }^{2}} In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes. The Hilbert scheme Hilb(n) of Pn classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme S, the set of S-valued points of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of Pn × S that are flat over S. The closed subschemes of Pn × S that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme Hilb(n) breaks up as a disjoint union of pieces Hilb(n, P) corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over Spec(Z). Grothendieck constructed the Hilbert scheme Hilb(n)S of n-dimensional projective space over a Noetherian scheme S as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme T over S, it represents the functor whose T-valued points are the closed subschemes of Pn ×S T that are flat over T. If X is a subscheme of n-dimensional projective space, then X corresponds to a graded ideal IX of the polynomial ring S in n + 1 variables, with graded pieces IX(m). For sufficiently large m, depending only on the Hilbert polynomial P of X, all higher cohomology groups of X with coefficients in O(m) vanish, so in particular IX(m) has dimension Q(m) − P(m), where Q is the Hilbert polynomial of projective space. Pick a sufficiently large value of m. The (Q(m) − P(m))-dimensional space IX(m) is a subspace of the Q(m)-dimensional space S(m), so represents a point of the Grassmannian Gr(Q(m) − P(m), Q(m)). This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial P into this Grassmannian. It remains to describe the scheme structure on this image, in other words to describe enough elements for the ideal corresponding to it. Enough such elements are given by the conditions that the map IX(m) ⊗ S(k) → S(k + m) has rank at most dim(IX(k + m)) for all positive k, which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take k = 1.) The Hilbert scheme Hilb(X)S is defined and constructed for any projective scheme X in a similar way. Informally, its points correspond to closed subschemes of X. Macaulay (1927) determined for which polynomials the Hilbert scheme Hilb(n, P) is non-empty, and Hartshorne (1966) showed that if Hilb(n, P) is non-empty then it is linearly connected. So two subschemes of projective space are in the same connected component of the Hilbert scheme if and only if they have the same Hilbert polynomial.