Fact-checked by Grok 2 weeks ago

Hilbert scheme

The Hilbert scheme is a fundamental construct in algebraic geometry that parametrizes closed subschemes of a projective scheme X over a base scheme S, classified according to a fixed Hilbert polynomial P; formally, it is the scheme \Hilb_P(X/S) representing the contravariant functor on the category of S-schemes that sends an S-scheme Y to the set of closed subschemes Z \subset X \times_S Y which are proper and flat over Y with Hilbert polynomial P on fibers.^[1] This functor is representable by a projective scheme over S.^[2] Introduced by Alexander Grothendieck in the early 1960s as part of his foundational development of scheme theory, the Hilbert scheme generalizes classical parameter spaces like the Grassmannian and provides a universal framework for studying families of subschemes and their deformations. The construction relies on techniques such as m-regularity and embedding into Grassmannians to establish existence and projectivity, with the tangent space at a point corresponding to a subscheme Z being isomorphic to the space of homomorphisms from the ideal sheaf conormal bundle to the structure sheaf of Z.^[2] Grothendieck's approach, detailed in Éléments de géométrie algébrique (EGA IV), built on earlier ideas from enumerative geometry and syzygy theory associated with David Hilbert's work on Hilbert polynomials, though the scheme-theoretic version marked a major advance in handling non-reduced structures and base change.^[3] The Hilbert scheme's importance lies in its role as a building block for moduli problems, such as constructing spaces of stable curves or vector bundles by embedding them as loci within suitable Hilbert schemes; for instance, the Hilbert scheme of points on a smooth projective surface is smooth and irreducible of dimension $2d for d points.^[1] It is connected when parametrizing subschemes of projective space, ensuring a unified parameter space despite the complexity of the objects involved.^[4] Applications extend to intersection theory, where it facilitates computations via the Grothendieck-Riemann-Roch theorem, and to derived categories, influencing modern areas like Donaldson-Thomas invariants in string theory and mirror symmetry.^[5]

Definition and Fundamentals

Functorial Definition

The concept of the Hilbert scheme originated in David Hilbert's 1890 work on invariant theory, where he sought canonical forms for binary forms by parametrizing ideals in polynomial rings, laying the groundwork for enumerating algebraic structures via what became known as the Hilbert polynomial.^[6] This classical motivation arose from the need to organize infinite families of invariants into finite-dimensional parameter spaces, addressing limitations in earlier approaches like those of Gordan.^[7] A key extension was the desire to compactify parameter spaces for subschemes, such as the Chow variety, which parametrizes effective cycles of fixed dimension and degree but fails to capture non-reduced or non-equidimensional structures in a complete manner.^[8] In the 1960s, Alexander Grothendieck generalized Hilbert's ideas within the framework of scheme theory, defining the Hilbert scheme as a representable functor that parametrizes flat families of subschemes, thus providing a universal moduli space for projective geometry over arbitrary bases.^[8] This functorial perspective shifted the focus from classical varieties to relative schemes, enabling the study of deformations and families in a categorical setting. The Hilbert functor \Hilb_{P(t)}^X is defined on the category of schemes over a base scheme S, where X/S is projective and P(t) \in \mathbb{Q} is a fixed polynomial. For any S-scheme T, the set \Hilb_{P(t)}^X(T) consists of isomorphism classes of pairs (Z, i), where i: Z \hookrightarrow X_T = X \times_S T is a closed immersion such that Z \to T is of finite presentation, flat, and proper, and the structure sheaf \mathcal{O}_Z satisfies the condition that its Hilbert function h_Z(d) = \chi(X_T, \mathcal{O}_Z(d)) equals P(d) for all sufficiently large integers d.^[1] The flatness of Z \to T ensures that the arithmetic genus and other cohomological invariants, including the Hilbert polynomial, are constant across fibers, allowing the functor to capture continuous families of subschemes with uniform topological type.^[8] Morphisms of schemes over S act contravariantly on the functor by base change, pulling back families via the fiber product.^[1] This setup establishes \Hilb_{P(t)}^X as a covariant functor from (\Sch/S)^{\op} to sets, with the fixed polynomial P(t) distinguishing components that parametrize subschemes of the same "size" in terms of leading cohomology dimensions, as opposed to varying degrees or genera.^[9] Grothendieck proved that this functor is representable by a projective scheme over S, providing a geometric object that universally parametrizes such families.^[8]

Hilbert Polynomial

The Hilbert polynomial is a fundamental invariant associated to a coherent sheaf on a projective scheme, capturing asymptotic growth information about its cohomology. For a coherent sheaf \mathcal{F} on a projective scheme X over a field k, the Hilbert function is defined as h_{\mathcal{F}}(m) = \chi(X, \mathcal{F} \otimes \mathcal{O}_X(m)), where \chi denotes the Euler characteristic and \mathcal{O}_X(m) is the m-th power of a fixed ample line bundle on X.^[10]^[11] There exists a unique polynomial P_{\mathcal{F}}(t) \in \mathbb{Q} of degree equal to the dimension of the support of \mathcal{F} such that h_{\mathcal{F}}(m) = P_{\mathcal{F}}(m) for all sufficiently large integers m \gg 0.^[10]^[11] This polynomial, defined with respect to a fixed ample line bundle, provides key geometric data, such as the dimension, degree (with respect to that line bundle), and arithmetic genus of the support.^[10] In the classical setting of a subscheme Z \subset \mathbb{P}^n_k defined by a saturated ideal, with structure sheaf \mathcal{O}_Z, the Hilbert polynomial takes the form

P_{\mathcal{O}_Z}(t) = \frac{\deg Z}{d!} t^d + \text{lower-degree terms},

where d = \dim Z and \deg Z is the degree of Z with respect to the hyperplane class.^[10]^[11] The leading coefficient determines the degree of Z, while lower terms encode refined invariants like the genus for curves.^[10] Explicit computations illustrate these features. For a zero-dimensional subscheme consisting of r points (counted with multiplicity) in \mathbb{P}^n, the Hilbert polynomial is the constant P(t) = r.^[11] For an integral curve C \subset \mathbb{P}^n of degree \delta and genus g, the polynomial is linear: P(t) = \delta t + (1 - g).^[10]^[11] The Hilbert polynomial exhibits additivity: if \mathcal{F} \oplus \mathcal{G} is a direct sum of coherent sheaves on X, then P_{\mathcal{F} \oplus \mathcal{G}}(t) = P_{\mathcal{F}}(t) + P_{\mathcal{G}}(t), reflecting the additivity of the Euler characteristic.^[10]^[11] More generally, additivity holds for short exact sequences of sheaves.^[10] By the Riemann-Roch theorem, for a curve the Hilbert polynomial equals the Euler characteristic \chi(C, \mathcal{O}_C(t)), which yields the explicit form \delta t + (1 - g) without further computation here.^[10]^[11] In the context of the Hilbert scheme, fixing a Hilbert polynomial specifies the component parametrizing subschemes with that invariant.^[11]

Construction in Projective Space

Determinantal Variety Approach

The construction of the Hilbert scheme \Hilb_{P(t)}^{\mathbb{P}^n} embeds it as a closed subscheme of a Grassmannian that parametrizes subspaces corresponding to quotients of global sections associated to subschemes of \mathbb{P}^n. Specifically, choose m sufficiently large so that the Hilbert function of any subscheme with Hilbert polynomial P(t) agrees with P(t) for all degrees \geq m; such an m exists by properties of the Hilbert polynomial. Let V = H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)), with \dim V = \binom{n+m}{m}, and let r = P(m). The relevant Grassmannian is \Gr(r, V), which parametrizes r-dimensional subspaces of V. A point in this Grassmannian corresponds to a quotient V \twoheadrightarrow Q \to 0 (or dually, a subspace U \subset V of dimension r) representing the space of global sections H^0(\mathcal{O}_Z(m)) for a potential subscheme Z \subseteq \mathbb{P}^n.^[2]^[12] This construction originates in Grothendieck's Éléments de géométrie algébrique (EGA IV), embedding the Hilbert scheme as a projective scheme via determinantal ideals in a Grassmannian.^[13] To enforce the conditions that the quotient defines a flat family with the correct Hilbert polynomial, consider the surjective map \phi: V \to Q, where Q represents the module of global sections of the quotient sheaf \mathcal{O}_Z(m) for a subscheme Z \subseteq \mathbb{P}^n. The associated sheafification must yield a coherent quotient \mathcal{O}_{\mathbb{P}^n} \twoheadrightarrow \mathcal{F} \to 0 with \chi(\mathcal{F}) = P and \mathcal{F}(m) globally generated. This map is represented by a matrix in the coordinates of the Grassmannian. The determinantal ideal defining the Hilbert scheme is generated by the (\dim V - r + 1) \times (\dim V - r + 1) minors of this matrix; the vanishing of these minors ensures that \phi is surjective and the associated graded pieces satisfy the Hilbert function conditions, corresponding to flat families with the fixed Hilbert polynomial.^[2]^[12] The key theorem states that \Hilb_{P(t)}^{\mathbb{P}^n} is precisely the zero locus of this determinantal ideal in the Grassmannian, hence a determinantal variety. Since the Grassmannian is projective over the base field, the Hilbert scheme inherits projectivity from this embedding. This construction, due to Grothendieck, provides an explicit algebraic realization using classical determinantal techniques.^[2]^[14] For a concrete example, consider \Hilb_1^{\mathbb{P}^2}, which parametrizes 0-dimensional subschemes of length 1 (i.e., points) in \mathbb{P}^2. Taking m=1, the Grassmannian is \Gr(1, H^0(\mathbb{P}^2, \mathcal{O}(1))) = \Gr(1, k^3) \cong \mathbb{P}^2, parametrizing 1-dimensional quotients of linear forms. The determinantal conditions are trivial in this case, yielding \Hilb_1^{\mathbb{P}^2} \cong \mathbb{P}^2 itself, where each point corresponds to the ideal generated by two independent linear forms vanishing at that point.^[2]

Quotient Bundle Construction

The quotient bundle construction realizes the Hilbert scheme as a special case of the Quot scheme, parametrizing coherent quotients of the structure sheaf \mathcal{O}_{\mathbb{P}^n}, leveraging the geometry of Grassmannians to ensure scheme-theoretic coherence. This approach, originally due to Grothendieck, emphasizes the functorial nature of quotients and facilitates extensions to families over arbitrary bases.^[14] Consider a finite-dimensional vector space V over an algebraically closed field k with \mathbb{P}^n = \mathbb{P}(V), and let \mathrm{Gr}(r, W) denote the Grassmannian scheme parametrizing rank-r quotients W \otimes \mathcal{O} \twoheadrightarrow Q \to 0, equipped with the tautological universal quotient bundle Q of rank r on \mathrm{Gr}(r, W). To construct the Hilbert scheme \Hilb^P_{\mathbb{P}(V)/k} parametrizing closed subschemes Z \subset \mathbb{P}(V) with Hilbert polynomial P, select an integer m \gg 0 such that H^i(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)}(m) \otimes \mathcal{I}_Z) = 0 for i > 0 and all such Z, by Castelnuovo-Mumford regularity. The construction proceeds by considering the Quot scheme \Quot_P(\mathcal{O}_{\mathbb{P}(V)} / \mathbb{P}(V) / k), which maps to the Grassmannian \mathrm{Gr}(P(m), H^0(\mathbb{P}(V), \mathcal{O}(m))) by sending a quotient \mathcal{O}_{\mathbb{P}(V)} \twoheadrightarrow \mathcal{F} \to 0 to the induced surjection on global sections H^0(\mathcal{O}(m)) \twoheadrightarrow H^0(\mathcal{F}(m)), well-defined for m \gg 0. The Hilbert scheme arises as the closed subscheme (flattening locus) where this quotient sheaf \mathcal{F} is flat over the base with constant Hilbert polynomial P, cut out by the vanishing of appropriate minors of the evaluation map, yielding a projective scheme.^[14]^[15] The universal quotient bundle Q on the Grassmannian induces relative quotients on the fibers, and the fixed-rank condition on the pushforward of the twisted structure sheaf guarantees that the associated subschemes are flat over the base with constant Hilbert polynomial, thus representing the desired functor. For the specific case of projective space \mathbb{P}^n = \mathbb{P}(V), this quotient construction yields a scheme isomorphic to the one obtained via the determinantal variety approach, with an explicit identification for subschemes supported in low degrees (e.g., zero-dimensional schemes).^[14] A key advantage of this construction lies in its inherent generality to relative Hilbert schemes over an arbitrary noetherian base scheme S. By replacing the Grassmannian with a relative Grassmannian bundle over S and extending the universal quotient to a relative setting on \mathbb{P}(E_S) \to S, the locus defined by the fixed-rank condition on the pushforward produces a projective scheme over S parametrizing flat families of subschemes with fixed Hilbert polynomial, without relying on global generation of ideals. This relative formulation underpins applications in moduli theory, such as deformation spaces of curves or sheaves.^[14]^[16]

Core Properties

Representability and Universality

Grothendieck established the representability of the Hilbert functor in a seminal result, showing that for a projective scheme X over \Spec k with k an algebraically closed field, the functor \Hilb_{P(t)}^X is represented by a projective scheme H. This functor assigns to any scheme S the set of closed subschemes Z \hookrightarrow X \times_k S which are flat and proper over S whose fibers have Hilbert polynomial P(t). The representing scheme H, known as the Hilbert scheme, is projective over \Spec k, guaranteeing its existence as a scheme and enabling the study of moduli problems in algebraic geometry.^[14] The universality of the Hilbert scheme H provides a canonical parameter space: it is equipped with a universal closed subscheme Z \hookrightarrow X \times_k H whose fibers over points of H realize all subschemes with Hilbert polynomial P(t). For any flat family Z' \hookrightarrow X \times_k S of such subschemes, there exists a unique morphism \phi: S \to H such that Z' is the base change Z \times_H S via \phi. This universal property, akin to the Yoneda lemma for schemes, ensures that H classifies all such families uniquely, making it the fine moduli space when it exists.^[14] The proof of representability relies on the construction of the dual Quot scheme, which parametrizes flat quotients of a fixed coherent sheaf on X. Specifically, for the structure sheaf \mathcal{O}_X, the Hilbert scheme embeds as an open subscheme of the Quot scheme \Quot_{\mathcal{O}_X}^{P(t)} via the correspondence between subschemes and their ideal sheaves or cokernels. Pro-representability is first established by approximating the functor through successive Grassmannians of quotients \Grass(r, H^0(X, \mathcal{O}_X(m))) for large m, using Castelnuovo-Mumford regularity to bound cohomology and ensure flatness via flattening stratifications. The transition maps between these Grassmannians are shown to be representable and proper, yielding a projective limit that represents the functor; projectivity of H then follows from the boundedness imposed by the fixed Hilbert polynomial and the properness of the universal family.^[14]

Tangent Space and Dimension

The tangent space to the Hilbert scheme \mathrm{Hilb}_P(X) at a point [Z] corresponding to a closed subscheme Z \subset X with ideal sheaf I_Z is isomorphic to \Ext^1_X(O_Z, I_Z). This identification arises from the cotangent complex of the Hilbert scheme, which controls the infinitesimal deformations of the exact sequence $0 \to I_Z \to O_X \to O_Z \to 0. Infinitesimal deformations of Z as an embedded subscheme correspond to elements of this Ext group, reflecting extensions of the defining sequence in the derived category. The obstruction space for lifting these first-order deformations to higher order is \Ext^2_X(O_Z, I_Z). Vanishing of this group ensures that the point [Z] is smooth in the Hilbert scheme, with the local structure determined by the tangent space. In general, the dimension of the tangent space provides an upper bound on the local dimension of the scheme at [Z]. The expected dimension of the Hilbert scheme at [Z] is given by \dim T_{[Z],\mathrm{Hilb}_P(X)} = \chi(X, N_{Z/X}), where N_{Z/X} = \Hom_X(I_Z, O_Z) is the normal sheaf of Z in X. This Euler characteristic equals h^0(X, N_{Z/X}) - h^1(X, N_{Z/X}), with the first term corresponding to the dimension of the tangent space when higher cohomology vanishes, and the second term accounting for potential obstructions or deficits in the embedding. The formula follows from the long exact sequence in Ext groups derived from the defining short exact sequence of sheaves, combined with Serre duality on the projective scheme X. For example, consider a smooth curve Z \subset \mathbb{P}^3 of degree \delta. The normal sheaf N_{Z/\mathbb{P}^3} splits as a sum of line bundles, and the expected dimension of the Hilbert scheme at [Z] is $4\delta, adjusted by the genus term (1 - g) in the full computation of \chi(N_{Z/\mathbb{P}^3}).^[17] This reflects the 4-dimensional freedom in deforming a line in \mathbb{P}^3, scaled by the degree, with smoothness ensuring the actual dimension matches the expected value.

Smoothness for Complete Intersections

In algebraic geometry, a closed subscheme Z \subset X of codimension c, where X is a smooth projective variety, is called a complete intersection if its ideal sheaf \mathcal{I}_Z is generated by a regular sequence f_1, \dots, f_c of global sections. For such Z, the point [Z] in the Hilbert scheme \Hilb_{P}(X), where P is the Hilbert polynomial of Z, is smooth. This follows from the vanishing of the obstruction space \Ext^2(\mathcal{O}_Z, \mathcal{I}_Z) = 0. The proof relies on the Koszul resolution of \mathcal{O}_Z, which is exact because f_1, \dots, f_c form a regular sequence:

$0 \to \bigwedge^c \mathcal{O}_X(-d_i) \to \cdots \to \bigwedge^1 \mathcal{O}_X(-d_i) \to \mathcal{O}_X \to \mathcal{O}_Z \to 0,

where d_i = \deg f_i. Applying \Hom(\cdot, \mathcal{I}_Z) to this resolution and taking cohomology yields the \Ext groups. Since \mathcal{I}_Z is generated by the regular sequence, the higher cohomology terms in degrees greater than 1 vanish, implying \Ext^2(\mathcal{O}_Z, \mathcal{I}_Z) = 0. This confirms that deformations of Z are unobstructed at [Z]. Moreover, the dimension of the Hilbert scheme at [Z] matches the expected dimension given by the Euler characteristic of the normal sheaf \chi(N_{Z/X}), where N_{Z/X} = \Ext^1(\mathcal{O}_Z, \mathcal{I}_Z). For the special case of hypersurfaces (i.e., c=1), this simplifies to \chi(N_{Z/X}) = \deg f_1 \cdot \deg Z + \dim H^0(\mathcal{O}_Z) - \dim H^1(\mathcal{O}_Z), aligning with the deformation space dimension. In general, complete intersections thus provide smooth points of the correct dimension in the Hilbert scheme. As a corollary, complete intersection subschemes admit unobstructed deformations, in contrast to general curves in projective space, where \Ext^2(\mathcal{O}_Z, \mathcal{I}_Z) may be nonzero, leading to potential singularities in the Hilbert scheme. This highlights the role of complete intersections as "nice" points in moduli problems.

Functorial and Relative Aspects

Representability for Projective Morphisms

In the relative setting, consider a projective morphism f: X \to S of finite type between noetherian schemes, equipped with a relatively ample line bundle \mathcal{O}_X(1). The relative Hilbert functor \Hilb_{P(t)}^{X/S} assigns to each S-scheme T the set of flat over T closed subschemes Z \subset X_T (where X_T = X \times_S T) such that Z is of finite presentation over T and each fiber of Z \to T has Hilbert polynomial the fixed polynomial P(t) \in \mathbb{Q}. A fundamental result establishes that this functor is representable by a proper scheme over S, often denoted \Hilb_{P(t)}^{X/S}. The proof proceeds via the associated relative Quot functor, which parametrizes S-flat quotients of \pi^* \mathcal{O}_X (where \pi: X_T \to X) by coherent sheaves of fixed type, with the Hilbert scheme arising as the special case where the kernel defines the structure sheaf of Z. For sufficiently large d, the Quot functor maps to the relative Grassmannian \Grass_S(r, f_* \mathcal{O}_X(d)) over S, which represents quotients of the pushforward f_* \mathcal{O}_X(d) by rank-r subsheaves; the desired locus is then cut out as a locally closed subscheme via conditions on the kernel sheaf being flat and the quotient supported properly. This construction leverages Castelnuovo-Mumford regularity to ensure finiteness and uses the projectivity of the Grassmannian to guarantee properness.^[15] The conditions for representability require f to be projective (hence proper and separated) and of finite type, ensuring that fibers are projective schemes and families behave well under base change. If S is an algebraic space rather than a scheme, the relative Hilbert functor is representable by a proper algebraic space over S, extending the result via Artin's criteria for algebraic stacks. When S = \Spec k for an algebraically closed field k, this recovers the classical absolute Hilbert scheme as a special case. This relative representability enables a uniform construction of Hilbert schemes over \Spec \mathbb{Z}, providing a foundation for arithmetic geometry by parametrizing families of subschemes with integral coefficients and fixed Hilbert polynomial, independent of the base field.

Relative Hilbert Schemes

In the setting of a morphism of algebraic spaces \phi: X \to S, the relative Hilbert functor \mathrm{Hilb}^P_{X/S} is defined as the functor that, to any scheme T \to S, associates the set of flat closed immersions Z \hookrightarrow X_T such that Z \to T is of finite presentation and the Hilbert polynomial of Z relative to \phi (computed using a relatively ample line bundle on X/S) is the fixed polynomial P. This parametrizes families of closed subschemes of X that are flat over the base and satisfy the cohomological condition encoded by P, generalizing the classical Hilbert scheme to relative situations over arbitrary bases. The representability of this functor depends on the nature of \phi. If X \to S is of finite presentation and separated, then \mathrm{Hilb}^P_{X/S} is representable by an algebraic space locally of finite presentation over S. Full representability by a scheme holds when \phi is projective, in which case the relative Hilbert scheme is proper over S. This extends the classical case where S = \mathrm{Spec}(k) and X is projective space, and aligns with representability results for projective morphisms as a special instance.^[9] A key advancement in this framework is due to A. J. de Jong, who employed Artin's approximation theorem adapted to algebraic spaces to establish these representability results, thereby extending Grothendieck's original theorem on the existence of Hilbert schemes from schemes to the broader category of algebraic spaces. This approach leverages formal versal deformations and their algebraization to construct the representing objects without relying on projectivity in the general case.^[18] In situations where the conditions for representability by an algebraic space fail—such as when the morphism \phi lacks finite presentation—the relative Hilbert functor may instead be algebraic but realized as a Deligne-Mumford stack, capturing the additional automorphisms or stacky structure arising from non-separated or more general geometric inputs.^[19]

Examples

Hilbert Scheme of Points

The Hilbert scheme of points on a scheme X, denoted \Hilb_r(X), parametrizes the zero-dimensional subschemes Z \subset X of length r, meaning that the structure sheaf \mathcal{O}_Z satisfies \dim_k H^0(Z, \mathcal{O}_Z) = r for an algebraically closed field k of characteristic zero, or equivalently, the saturated ideals I_Z \subset \mathcal{O}_X with \dim_k \mathcal{O}_X / I_Z = r.^[5] This construction applies generally to schemes X, but is particularly well-studied for X = \mathbb{P}^n or smooth projective surfaces, where the Hilbert polynomial is the constant function P(t) = r.^[1] For a smooth projective surface X, \Hilb_r(X) is a smooth, irreducible, projective variety of dimension $2r.^[5] In the case of X = \mathbb{P}^2, \Hilb_r^{\mathrm{red}}(\mathbb{P}^2) specifically parametrizes the zero-dimensional subschemes of degree r, inheriting the smoothness and irreducibility properties.^[20] The variety \Hilb_r(X) compactifies the configuration space of r points on X by including non-reduced structures, such as multiple points or infinitesimal thickenings. For r = 1, \Hilb_1(X) is isomorphic to X itself, as the only zero-dimensional subschemes of length 1 are the reduced points.^[1] For small r, such as r = 2, \Hilb_r(X) can be described as a blow-up of the symmetric product \Sym^r(X) along the locus of non-reduced schemes, resulting in a structure that is a \mathbb{P}^1-bundle over the smooth part of \Sym^r(X).^[5] In general, the Hilbert-Chow morphism \pi: \Hilb_r(X) \to \Sym^r(X) provides a crepant resolution of the singularities of the symmetric product, where \pi contracts exceptional \mathbb{P}^1-divisors over coincident points.^[1] This resolution property makes \Hilb_r(X) a key tool in enumerative geometry, as it regularizes the symmetric product for intersection theory computations.^[5] For X = \mathbb{P}^2, \Hilb_r(\mathbb{P}^2) features prominently in Götzsche's conjecture, which predicts explicit formulas for the refined Poincaré polynomials (or Betti numbers with multiplicities) of these schemes, later proved using wall-crossing techniques in Donaldson-Thomas theory.^[21]

Fano Schemes

The Fano scheme F_k(V) of a hypersurface V \subset \mathbb{P}^n of degree d is defined as the closed subscheme of the Hilbert scheme \mathrm{Hilb}^{\mathbb{P}^n} consisting of all subschemes \Lambda \cong \mathbb{P}^k contained in V.^[22] Equivalently, it is the locus in the Grassmannian \mathrm{Gr}(k+1, n+1) parametrizing (k+1)-dimensional linear subspaces of \mathbb{P}^n that lie entirely in V.^[22] This locus arises naturally as the support of subschemes with Hilbert polynomial p(t) = \binom{t + k}{k}, making F_k(V) a component of the corresponding Hilbert scheme.^[1] The structure of F_k(V) is that of a determinantal variety: it is the zero locus in the Grassmannian of a global section of the vector bundle whose fiber over a point corresponding to \Lambda is H^0(\mathcal{O}_\Lambda(d)) \cong \mathrm{Sym}^d(S^\vee), where S is the tautological subbundle on the Grassmannian.^[23] This construction arises from the condition that the defining equation of V restricts to zero on \Lambda, equivalent to the vanishing of the image of this section.^[22] If non-empty, F_k(V) is smooth of the expected dimension (n - k)(k + 1) - \binom{k + d}{d}.^[24] A classical example is the case of lines (k=1) on a smooth cubic surface (n=3, d=3) in \mathbb{P}^3: here F_1(V) consists of exactly 27 points over an algebraically closed field.^[22] For a smooth cubic threefold (n=4, d=3) in \mathbb{P}^4, F_1(V) is a smooth surface of expected dimension 2 with irregularity q=0.^[22]

Hypersurfaces of Fixed Degree

The Hilbert scheme \operatorname{Hilb}_{P(t)}(\mathbb{P}^n), where P(t) is the Hilbert polynomial of a degree-d hypersurface in \mathbb{P}^n, has a distinguished component that parametrizes closed subschemes with this fixed polynomial. This component contains an open subset consisting of the degree-d hypersurfaces, which are the scheme-theoretic zero loci of homogeneous polynomials of degree d. This open set is isomorphic to the projective space \mathbb{P}^N, where N = \binom{n+d}{d} - 1, corresponding to the projective space of coefficients of such polynomials modulo scalars.^[1] The full component includes, in its closure, non-reduced schemes such as multiple structures on lower-degree hypersurfaces, as well as potentially non-hypersurface schemes that limit to these configurations while preserving the Hilbert polynomial P(t). For instance, degenerations can yield schemes supported on unions of lower-dimensional components with imposed structure, all sharing the same P(t). The expected dimension of this component is \binom{n+d}{d} - 1, achieved at points corresponding to reduced hypersurfaces, where the scheme is smooth.^[1] A concrete example arises with quadric hypersurfaces (d=2) in \mathbb{P}^3, where the relevant Hilbert scheme component is \mathbb{P}^9, parametrizing all quadrics via symmetric $4 \times 4 matrices up to scalar. The locus of singular quadric hypersurfaces within this space corresponds to rank-deficient matrices (rank at most 3), forming a determinantal hypersurface of degree 4 in \mathbb{P}^9. This locus itself exhibits singularities along the further degenerate cases of corank greater than 1.^[1]

Curves and Moduli Spaces

The Hilbert scheme \Hilb_{\delta t + 1 - g}(\mathbb{P}^n) parametrizes subschemes of \mathbb{P}^n that are pure one-dimensional of degree \delta and arithmetic genus g, with the Hilbert polynomial \delta t + 1 - g.^[25] The main irreducible component of this scheme is the closure of the locus parametrizing smooth curves embedded in \mathbb{P}^n via a complete linear series of degree \delta.^[25] This component provides a natural compactification of the space of smooth embeddings, incorporating limits such as nodal or multiple curves while preserving the Hilbert polynomial.^[26] For pointed curves, the Hilbert scheme of n-pointed curves of genus g and degree \delta in \mathbb{P}^n admits a morphism to the Deligne-Mumford compactification \overline{\mathcal{M}}_{g,n} by forgetting the embedding and marking the points.^[27] An intermediate space is the Kontsevich moduli space of stable maps \overline{\mathcal{M}}_{g,n}(\mathbb{P}^n, \delta), which receives a map from the pointed Hilbert scheme by associating to each embedded pointed curve its tautological map from the curve to \mathbb{P}^n.^[28] This chain of maps connects embedded compactifications to abstract curve moduli, facilitating enumerative invariants and birational studies.^[28] A concrete example is the Hilbert scheme \Hilb_{3t}(\mathbb{P}^2) of plane cubics, which parametrizes subschemes of degree 3 and genus 1 in the plane. This component is isomorphic to \mathbb{P}^9 minus the discriminant locus, where \mathbb{P}^9 parametrizes all ternary cubics via the projectivization of the space of homogeneous cubics in three variables.^[25] The discriminant is the hypersurface of singular cubics, defined by a degree-12 invariant vanishing on reducible or cuspidal curves.^[25] For space curves in \mathbb{P}^3, the main component of the Hilbert scheme of degree \delta curves has dimension $4\delta.^[17] For high genus, the Hilbert scheme \Hilb_{(2g-2)t + 1 - g}(\mathbb{P}^{g-1}) of canonically embedded curves often exhibits multiple irreducible components, arising from curves with special linear series or different embedding types.^[29] The principal component, parametrizing general canonical curves, is birational to \overline{\mathcal{M}}_g via the map sending a curve to its canonical model, with the inverse obtained through Veronese re-embeddings of rational normal curves in limits.^[27] These multiple components highlight challenges in uniform compactification, as lower-dimensional loci may dominate in special cases.^[29]

Advanced Topics

Hilbert Schemes on Manifolds

The Hilbert scheme of points on a complex manifold M, denoted \mathrm{Hilb}_r(M), parametrizes zero-dimensional subschemes of M of length r. For a smooth complex manifold M of complex dimension d, \mathrm{Hilb}_r(M) is a smooth complex manifold of complex dimension r d, hence a smooth real manifold of dimension $2 r d.^[30] This space is constructed in the analytic category using the deformation functor of ideal sheaves defining length-r subschemes, yielding a universal deformation space that is smooth under suitable conditions on M. Alternatively, \mathrm{Hilb}_r(M) serves as a desingularization of the symmetric product \mathrm{Sym}^r M = M^r / S_r, with the Hilbert-Chow morphism \mathrm{Hilb}_r(M) \to \mathrm{Sym}^r M being a resolution of singularities that is an isomorphism over the locus of r distinct reduced points.^[5] When M is Kähler, \mathrm{Hilb}_r(M) carries a natural holomorphic symplectic structure. This structure is induced by the Kähler form on M via the universal subscheme, leveraging Serre duality to define a non-degenerate closed holomorphic 2-form on the tangent bundle of \mathrm{Hilb}_r(M). A representative example arises when M = \mathbb{C}^n, where \mathrm{Hilb}_r(\mathbb{C}^n) is smooth and open in the algebraic Hilbert scheme of points on \mathbb{P}^n; compactifying by the hyperplane at infinity recovers the full projective Hilbert scheme, with boundary components corresponding to subschemes supported at infinity.^[30]

Hyperkähler Geometry Connections

The Hilbert scheme of r points on a K3 surface, denoted \mathrm{Hilb}^r(S), is a hyperkähler manifold of dimension $2r.^[31] This structure arises from the holomorphic symplectic form induced on the scheme, making it irreducible holomorphic symplectic (IHS) and compact when S is projective.^[32] Moreover, \mathrm{Hilb}^r(S) serves as the prototype for the deformation type known as K3^{}-type hyperkähler manifolds, with all such deformations sharing the same topological and symplectic properties.^[31] Nakajima quiver varieties provide a geometric realization of Hilbert schemes of points on \mathbb{C}^n as hyperkähler quotients.^[33] Specifically, for the Jordan quiver and appropriate dimension vectors, the quiver variety is constructed from representations of a quiver, quotiented by the action of product groups using a hyperkähler moment map \mu = (\mu_\mathbb{C}, \mu_\mathbb{R}), where \mu_\mathbb{C} and \mu_\mathbb{R} take values in complex and real Lie algebras, respectively.^[34] This quotient yields \mathrm{Hilb}^r(\mathbb{C}^n) as a smooth hyperkähler manifold, embedding it within the broader framework of symplectic reductions that preserve the hyperkähler structure.^[33] For a finite subgroup \Gamma \subset \mathrm{SL}(2, \mathbb{C}), the \Gamma-equivariant Hilbert scheme of points on \mathbb{C}^2 furnishes the minimal hyperkähler resolution of the quotient singularity \mathbb{C}^2 / \Gamma.^[35] This resolution is smooth and crepant, with the exceptional locus corresponding to \Gamma-invariant subschemes, and it inherits a hyperkähler metric compatible with the quotient's asymptotic behavior. The resolved manifold carries the Beauville–Bogomolov–Fujiki (BBF) form, a quadratic form on the second cohomology that defines the Fujiki relations and governs the manifold's period map.^[31] These hyperkähler Hilbert schemes underpin applications to Donaldson-Thomas (DT) invariants, computed via virtual fundamental classes on the schemes parameterizing stable sheaves.^[36] In the hyperkähler setting, such as for 4-fold resolutions, the virtual classes refine DT invariants by incorporating modified obstruction theories, leading to vanishing results and connections to BPS counts on Calabi–Yau varieties.^[37] This framework extends DT theory beyond 3-folds, enabling enumerative invariants that encode hyperkähler geometry through wall-crossing phenomena.^[36]

References

[1]
[PDF] the hilbert scheme
The Hilbert scheme parameterizes subschemes of projective space with a fixed Hilbert polynomial, thus provides the starting point for all such constructions. We ...
[2]
[PDF] ELEMENTARY INTRODUCTION TO REPRESENTABLE ...
The important exception is the Hilbert scheme, which is a countable union of noetherian schemes, hence only locally noetherian. 2. Functors. The functor of ...
[3]
[PDF] Éléments de géométrie algébrique : IV. Étude locale des schémas et ...
Dec 8, 2012 · É.S. ALEXANDER GROTHENDIECK. Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas ...
[4]
[PDF] Connectedness of the Hilbert scheme - Numdam
In this paper we study the Hilbert scheme of subschemes of projective space, as defined by Grothendieck [FGA, p. 221-01 ft] (2). Let S be a noetherian prescheme ...Missing: original | Show results with:original
[5]
[PDF] HILBERT SCHEMES - UMD Math Department
It is an algebraic family of ideals and it defines a curve in the Hilbert scheme. The limit as t → 0 should be a length 2 ideal that is supported at the ...
[6]
[PDF] Joachim Jelisiejew – Hilbert schemes of points and applications
May 9, 2017 · The Hilbert scheme Hilb(X) was constructed by Grothendieck [Gro95]. It decomposes into a disjoint union of Hilbp(X), parameterized by the ...<|control11|><|separator|>
[7]
[PDF] David Hilbert and the theory of algebraic invariants
Hilbert, in his lectures, limits himself to binary forms but says that generalizing to m-ary forms poses no difficulties in most cases. He writes a general ...Missing: scheme | Show results with:scheme<|control11|><|separator|>
[8]
[PDF] Construction of Hilbert and Quot Schemes - arXiv
The original reference for the following argument is EGA IV (2) 2.8.1. The ... The scheme- theoretic analogue of the above is the following theorem of ...
[9]
Section 99.9 (0CZX): The Hilbert functor—The Stacks project
If f is of finite presentation and separated, then \mathrm{Hilb}_{X/B} is an algebraic space locally of finite presentation over B.Missing: Hilb_P( | Show results with:Hilb_P(
[10]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 31
Feb 14, 2006 · From the Hilbert polynomial, we can extract many invariants, of which two are par- ticularly important. The first is the degree. Classically, ...
[11]
[PDF] Math 259x: Moduli Spaces in Algebraic Geometry
Aug 26, 2020 · The Hilbert polynomial encodes a lot of geometric information about a projective ... Now we can define Hilbert functor of a projective morphism f ...
[12]
[PDF] 5 Hilbert schemes - Brown Math
Lemma 23 For each e ≥ d and R ∈ N, Hp(e, R) is a closed subscheme of the Grassmannian, with an explicit set of defining polynomial equations. proof: We give a ...
[13]
[PDF] Construction of Hilbert and Quot Schemes - arXiv
Abstract. This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk 'Techniques de construction et théor` ...
[14]
[PDF] Hilbert and Quot Schemes - UW Math Department
Nov 14, 2021 · The Grassmanian and the Hilbert scheme are special cases of the Quot scheme: GrS(k, V ) ∼= QuotP (V/S/S) where P(z) = k is the constant ...
[15]
[PDF] HILBERT AND QUOT SCHEMES
This almost finishes the story of the construction of the Hilbert scheme and even more, its generalization Quot scheme. However we should slightly modify the ...
[16]
Representability of Hilbert schemes and Hilbert stacks of points - arXiv
Feb 26, 2008 · We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space.
[17]
[PDF] on the dimension of the hilbert scheme of curves
... normal sheaf NX/Pn splits into L k i=1 OX(di). Therefore, in this case we can compute χ(NX/Pn |C) explicitly as χ k. M i=1. OC(di) = k. X i=1. (1 − g + ...
[18]
None
### Summary of Smoothness of Hilbert Schemes at Complete Intersection Points
[19]
[PDF] approximation of versal deformations - Mathematics
BRIAN CONRAD AND A.J. DE JONG. 1. Introduction. In Artin's work on algebraic spaces and algebraic stacks [A2], [A3], a crucial ingredient is the use of his.
[20]
[PDF] artin's criteria for algebraicity revisited
Jun 20, 2012 · generalized Grothendieck's existence result and showed that the Hilbert and Pi- card schemes exist—as algebraic spaces—in great generality.
[21]
[PDF] Punctual Hilbert Schemes of the Plane - Harvard Math
Mar 19, 2018 · This paper discusses punctual Hilbert schemes, Hilbert schemes that parametrize collec- tions of points. When the points lie on a two ...
[22]
[1010.3211] A short proof of the Göttsche conjecture - arXiv
Oct 15, 2010 · The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of ...Missing: P^ | Show results with:P^
[23]
[PDF] Foundations of the theory of Fano schemes - Numdam
Let F denote the Fano scheme of lines on a cubic hypersurface X. over a field. The first main result (1.3) is that F is the scheme of zeroes.
[24]
[PDF] 3264 & All That Intersection Theory in Algebraic Geometry
We have been working on this project for over ten years, and at times we have felt that we have only brought on ourselves a plague of locus.
[25]
[PDF] Fano schemes of symmetric and alternating matrices of bounded rank
As an application, we provide geometric arguments for several previous results concerning spaces of symmetric and alternating matrices of bounded rank. Keywords ...Missing: O_V( | Show results with:O_V(
[26]
[PDF] The Hilbert schemes of degree three curves - Numdam
- H(3,l) parametrizes plane cubic curves, and is a P9 bundle over (P3)*, hence is irreducible of dimension 12. That ff(3,0) and H(3,-l) are smooth and.
[27]
[0808.3604] On the dimension of the Hilbert scheme of curves - arXiv
Aug 26, 2008 · We give lower bounds for the dimension of such a component when X is P^3, P^4 or a smooth quadric threefold in P^4 respectively.Missing: 6δ | Show results with:6δ
[28]
[PDF] The irreducibility of the space of curves of given genus
Fix an algebraically closed field k. Let M.g be the moduli space of curves of genus g over k. The main result of this note is that M.g is irreducible for ...
[29]
[PDF] the kontsevich moduli spaces of stable maps
Since a degree 0 map from a connected curve is determined by specifying a point ... The Kontsevich moduli space ad mits a morphism to the Deligne-Mumford moduli ...
[30]
On some components of Hilbert schemes of curves - arXiv
Jan 24, 2021 · Let \mathcal{I}_{d,g,R} be the union of irreducible components of the Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree d, ...
[31]
[PDF] Hilbert Schemes of Points on Surfaces - arXiv
Apr 21, 2003 · A Kähler manifold X of real dimension 4n is called hyperkähler if its holonomy group is Sp(n). Compact complex manifolds are holomorphic ...
[32]
[PDF] Teichmüller space via Kuranishi families - Numdam
A convenient reference for deformation theory and Hilbert schemes is [33]. A detailed presentation of Kuranishi families will be contained in the forthcoming.
[33]
[PDF] Hilbert schemes of K3 surfaces, generalized Kummer, and ... - arXiv
Oct 5, 2021 · There is a mysterious link (in fact related to mirror symmetry) between hyper-Kähler manifolds of dimension 2n and rational homology projective ...
[34]
https://homepage.mi-ras.ru/~akuznet/publications/QuiverVarietiesAndHilbertSchemes.pdf
[35]
[PDF] examples of hyperk¨ahler manifolds as moduli spaces of sheaves on ...
scheme of points on a K3 surface is a compact hyperkähler manifold. We will construct a holomorphic symplectic form on the Hilbert scheme, then we will show.<|control11|><|separator|>
[36]
https://people.math.ethz.ch/~rahul/dtlc.pdf
[37]
[PDF] Quiver varieties and Hilbert schemes
QUIVER VARIETIES AND HILBERT SCHEMES. 697. [10] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American ...
[38]
[PDF] noncompact complex symplectic and hyperk¨ahler manifolds ...
Hilbert Scheme of points on C2. ... Thus the underlying differentiable manifold is the minimal resolution of the rational double point (Kleinian singularity) C2/Γ ...
[39]
[PDF] hilbert schemes of points and the adhm construction
Overview. In this talk, I discuss a how a special case of the Instanton Moduli space on R4 can be described as a Hilbert Scheme of points on C2.
[40]
[PDF] The local Donaldson-Thomas theory of curves
The Hilbert scheme vanishing is obtained from the existence of a modified virtual class in the hyperkähler setting. The proof requires a restriction of the ...
[41]
[PDF] Holomorphic anomaly equations for the Hilbert scheme of points of a ...
Dec 2, 2024 · (2)). The rank 1 Donaldson-Thomas invariants of X in arbitrary curve classes are determined from the series DTn by the multiple cover ...