Hilbert Schemes of Points on Surfaces

• Hilbert schemes of points on surfaces are moduli spaces that parameterize zero-dimensional subschemes of fixed length n, exhibiting smooth, irreducible, and quasi-projective structures.
• Their cohomology, derived invariants, and Picard groups capture both the geometry of the underlying surface and combinatorial partition data, as evidenced by generating functions like Göttsche’s formula.
• They serve as pivotal examples in moduli theory, linking lattice theory, birational geometry, and enumerative invariants, with applications to hyperkähler manifolds and mathematical physics.

A Hilbert scheme of points on a surface is a fine moduli scheme parameterizing zero-dimensional subschemes of fixed length n on an algebraic surface S. Denoted S^[n], this space is a smooth, irreducible, quasi-projective variety of dimension 2n when S is smooth and connected. Hilbert schemes of points play a central role in algebraic geometry, encoding rich geometric, arithmetic, and representation-theoretic structures, and they serve as standard test objects in moduli theory, enumerative geometry, and theoretical physics.

1. Construction and Basic Properties

Given a smooth projective surface S over an algebraically closed field, the Hilbert scheme S^[n] parameterizes closed subschemes Z ⊆ S with length n. The Hilbert–Chow morphism associates to a subscheme Z its support cycle in the n-th symmetric product S^n. S^[n] is a smooth, irreducible, quasi-projective variety of dimension 2n. Fogarty's theorem generalizes to surfaces with at worst rational double point singularities: for any quasi-projective normal surface X with only rational double point singularities, Hilb^d(X) is irreducible of dimension 2d (Zheng, 2017).

The decomposition of the cohomology and Picard group of S^[n] is controlled by the geometry of S and the universal family. The Néron-Severi group decomposes as

$$\mathrm{Pic}(S^{[n]}) \cong \mathrm{Pic}(S) \oplus \mathbb{Z} \cdot \left(\tfrac{1}{2}B\right)$$

where B is the exceptional divisor of the Hilbert–Chow morphism (Bolognese et al., 2015). S^[n] is smooth and projective for projective S, and has trivial canonical bundle when S is a K3 surface.

For singular surfaces with rational double points, matrix factorization and maximal Cohen–Macaulay module theory show that Hilbert schemes remain irreducible, and smooth points correspond to ideals of finite projective dimension (Zheng, 2017).

2. Cohomological and Derived Invariants

The cohomology of S^[n] reflects both the geometry of S and the combinatorics of partitions. If H^*(S,ℤ) is torsion-free, then so is H^*(S^[n],ℤ) for all n, and S^[n] inherits trivial Chow motive when S does (Totaro, 2019). The generating function for Betti numbers is given by Göttsche's formula:

$$\sum_{n=0}^\infty p(S^{[n]}, t)q^n = \prod_{k=1}^\infty \prod_{j} (1 - (-t)^{2k-2+j} q^k)^{-(-1)^{j+1}b_j(S)}.$$

For surfaces with torsion cohomology, subtleties arise in determining the cohomology ring of S^[n].

The Hochschild cohomology of S^[n] is governed not only by the Hochschild cohomology of S, but by its full Hochschild–Serre cohomology. Concretely, for a smooth surface S,

$$\bigoplus_{n=0}^\infty \mathrm{hochserre}_k(S^{[n]}) \cdot t^n \cong \mathrm{Sym}^*\left( \bigoplus_{i\ge1}\mathrm{hochserre}_{1+(k-1)\cdot i}(S) \cdot t^i \right),$$

where $$\mathrm{hochserre}_k(X) = \mathrm{RHom}_{X\times X}(\Delta_* \mathcal{O}_X, \Delta_*(\omega_X^{\otimes k})[k \cdot d_X])$$ (Belmans et al., 2023). The deformation theory of S^[n] is not simply the symmetric power of that for S: the tangent space to deformations of S^[n] contains an extra summand $H^0(S,\omega_S^{-1})$ when $H^1(S, \mathcal{O}_S)=0$, matching the appearance of global Poisson structures (Belmans et al., 2023).

A generating function for twisted Hodge numbers of S^[n] with values in tautological line bundles L_n is given (correcting conjectures by Boissiére) by

$$\sum h^{p,q}(S^{[n]}, L_n) x^p y^q t^n = \prod_{k \ge 1}\prod_{p,q=0}^2 (1 - (-1)^{p+q} x^{p+k-1} y^{q+k-1} t^k)^{-(-1)^{p+q} h^{p,q}(S, L^{\otimes k})}$$

(Belmans et al., 2023).

3. Moduli, Lattice Theory, and Density in Moduli Spaces

Hilbert schemes S^[n] of K3 surfaces S give key examples of irreducible holomorphic symplectic (IHS) manifolds. The locus of marked pairs (X, η) with X ≅ S^[n] for some projective K3 S is dense in the moduli space of IHS manifolds of that deformation type. The density theorem is established via lattice-theoretic analysis, period domains,

$$\mathcal{D} = \{p \in \mathbb{P}(A \otimes \mathbb{C}) : (p, p) = 0, (p, \bar{p}) > 0\}$$

where A is the second cohomology lattice with Beauville–Bogomolov pairing, and monodromy group actions (Markman et al., 2011). Monodromy and parallel-transport isometries allow one to connect period points to Hilbert schemes S^[n] and generalized Kummer varieties. This density means that geometric and arithmetic properties of K3 surfaces are ubiquitous in the deformation space of IHS manifolds.

4. Birational and Automorphism Group Structures

Automorphisms of S^[n] are natural (induced from automorphisms of S), except in specific cases. For weak Fano or general type surfaces, all automorphisms of S^[n] are natural. The unique exception arises for X = C₁ × C₂ (product of curves), n = 2, where a non-natural automorphism is given by swapping points in the second coordinate (Belmans et al., 2019).

For abelian surfaces, there exist non-natural automorphisms of A^[n] which preserve the exceptional divisor of nonreduced subschemes, provided the Picard rank is at least 2; for generic (principally polarized, Picard rank 1) abelian surfaces, all automorphisms are natural (Girardet, 2 Jul 2024). The group of birational automorphisms of S^[n], for K3 S of Picard rank one, is classified via solutions to Pell's equations,

$X^2 - t(n-1)Y^2 = 1,$

and is connected to the reflection group of the BBF lattice; significant distinctions arise between symplectic and non-symplectic types, and between strictly birational versus biregular automorphisms (Beri et al., 2020).

Birational geometry also connects to effective and nef cone structures. Wall-crossing in Bridgeland stability and the Positivity Lemma of Bayer and Macrì are fundamental for describing the ample and nef cones of S^[n]. There exist explicit formulas for extremal nef divisors in terms of data from S, e.g.,

$D = H^{[n]} - \frac{a^2 H^2 - K_X \cdot H}{2}\, B$

in the Picard rank one case (Bolognese et al., 2015).

5. Intersections, Enumerative Geometry, and K-Theory

Intersection-theoretic invariants—such as generating functions for Euler characteristics, Segre classes, and tautological bundles—on S^[n] are controlled by universal formulas. Göttsche's formula relates the generating function

$$1 + \sum_{n>0} q^n \chi(S^{[n]}) = \prod_{n>0} (1 - q^n)^{-e(S)}$$

for a smooth surface S. The Carlsson–Okounkov formula computes generating functions for the Euler class of the twisted tangent bundle $T_{S^{[n]}}(M)$, and its relative version for ideals relative to a divisor C uses a product over the normal bundle and correction terms, as in

$Z_A(S/C, M) = Z_A(S, M) \cdot Z_A(N, M^*)$

(Gholampour et al., 2015).

The theory of equivariant homology and K-theoretic invariants is highly developed for (ℂ²)^[n]: the torus $(ℂ^*)^2$ acts with isolated fixed points parameterized by partitions; equivariant localization computes action of convolution operators, realizing a Fock space representation of the Heisenberg algebra and, via tautological bundles, a Virasoro action (Nakajima, 2014). K-theoretic descendent series (holomorphic Euler characteristics of tautological sheaves) are rational functions of explicit degree (Arbesfeld, 2022).

For toric surfaces, Newton–Okounkov bodies provide convex-geometric descriptions of asymptotics of linear series on S^[n] and yield upper bounds on the effective cone. The conjecture is that, for projective spaces and Hirzebruch surfaces, these bodies are sharp (Cavey, 2022).

6. Stratification, Brill–Noether Loci, and Deformation Theory

The stratification of S^[n] according to punctual or local conditions yields a detailed picture, particularly for the Hilbert–Samuel stratification of punctual Hilbert schemes; affine charts can be indexed by combinatorial data (such as Young diagrams or "Young walls" in singular settings) (Gyenge, 2016). Brill–Noether loci in S^[n] × S, parametrizing (I,p) such that I needs ≥ r+1 generators at p, are irreducible, non-empty (when expected dimension is positive), and Cohen–Macaulay, with expected codimension r(r+1) (Bayer et al., 2023).

In the context of Poisson surfaces, Hilbert schemes inherit Bottacin Poisson structures; symplectic groupoids integrate the induced brackets, and local models (via syzygies) identify open pieces with duals of Lie algebras of affine transformations. For surfaces with reduced, nodal anticanonical divisors, (S^[n], π_H) is holonomic, and the associated deformation theory exhibits finiteness properties (Matviichuk et al., 2022).

7. Birational Geometry, Motives, and Stable Birationality

Hilbert schemes S^[n] on rational or minimal geometrically rational surfaces are stably birationally periodic: for n large, Hilb^n_S is stably birational to Hilb^{n+\mathrm{ind}(S)}_S, where ind(S) is the index (the gcd of degrees of closed points) (Porzio, 14 Aug 2024). This stable periodicity implies rationality of the motivic zeta function for Hilbert schemes,

$$\zeta_{\mathrm{mot}}^{(\mathrm{Hilb})}(X, t) = 1 + \sum_{n=1}^\infty [\mathrm{Hilb}^n_X] t^n,$$

in the Grothendieck ring K_0(\mathrm{Var}/k)/(\mathbb{L}) over characteristic zero.

The connection between birational geometry and effective cones is further illuminated by weak Lefschetz-type theorems for Hilbert schemes. Under birational contractions, the effective cone and the structure of base loci (in particular, the augmented stable base locus decomposition) behaves well, and divisorial classes arising from Severi varieties (loci of n-nodal curves) often provide explicit extremal effective divisors (Huerta et al., 2018).

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In summary, Hilbert schemes of points on surfaces serve as a nexus for the interaction between moduli theory, lattice theory, representation theory, derived categories, enumerative geometry, and birational and motivic invariants. Their structure encodes not just the geometry of the underlying surface, but subtler derived and deformation-theoretic invariants, with implications for the paper of hyperkähler manifolds, moduli of sheaves, and mathematical physics.