Hilbert scheme

Hilbert scheme
Name	Hilbert scheme
Field	Algebraic geometry
Introduced	David Hilbert
Related	Scheme, Moduli space, Grassmannian

Hilbert scheme

The Hilbert scheme is a fundamental construction in algebraic geometry providing a parameter space for closed subschemes of a projective scheme with fixed Hilbert polynomial. It arose from work of David Hilbert and later constructions by Alexander Grothendieck and connects to many topics such as the Grassmannian, the Quot scheme, and the Picard scheme, with applications ranging through the work of Oscar Zariski, Jean-Pierre Serre, and Michael Artin.

Definition and basic properties

A Hilbert scheme parametrizes flat families of closed subschemes of a fixed projective scheme with a prescribed Hilbert polynomial; the construction is classically stated for a projective embedding given by a very ample line bundle on a base like Projective space, relying on cohomological input from results of Serre duality and techniques inspired by Noetherian ideas. The functor of points perspective used by Grothendieck and developed in his work with the Séminaire de Géométrie Algébrique shows representability by a projective scheme, relating to standard objects such as the Grassmannian and the Plücker embedding, and making contact with deformation-theoretic tools of Kodaira–Spencer type and criteria from Zariski and Nagata.

Examples and important cases

Key examples include the Hilbert scheme of points on Projective plane and on smooth projective surfaces such as a K3 surface or an Enriques surface, where the Hilbert scheme of n points often yields smooth varieties studied by Friedman, Beauville, and Mukai. The Hilbert scheme of curves in Projective 3-space is central to classical studies by Max Noether and modern investigations by Gruson, Peskine, and Hartshorne, while zero-dimensional Hilbert schemes on surfaces relate to the McKay correspondence explored by John McKay and generalized by Bridgeland, King, and Reid. Special loci such as the punctual Hilbert scheme and components studied by Fogarty illustrate pathologies and smoothness phenomena important for researchers like Gottsche and Haiman.

Construction and representability

Grothendieck's construction embeds the Hilbert functor into a suitable Grassmannian using boundedness results and cohomology vanishing theorems originated in work by Serre and systematized by Grothendieck in his theory of Schemes. Representability is proven by constructing a closed subscheme of a Grassmannian via determinantal conditions related to Castelnuovo–Mumford regularity and results attributable to Castelnuovo and Mumford. Techniques from the theory of the Quot scheme and flattening stratification owing to Raynaud and Gruson are essential, and deformation-obstruction calculations use the formalism established by Schlessinger and Illusie.

Geometry and topology of Hilbert schemes

The geometry of Hilbert schemes exhibits rich structure: smoothness criteria for the Hilbert scheme of points on surfaces were established by Fogarty, while symplectic structures on Hilbert schemes of points on K3 surfaces were studied by Beauville and Fujiki. Cohomological descriptions employ tools from Hodge theory developed by Deligne and intersection theoretic methods from Mumford, with connections to perverse sheaves as in work by Beilinson, Bernstein, and Deligne. Topological invariants such as Betti numbers and generating functions were computed in seminal work by Göttsche and play roles in comparisons with invariants from Donaldson theory and Seiberg–Witten theory studied by Witten and Donaldson.

Connections to moduli problems and deformation theory

Hilbert schemes provide fine or coarse moduli spaces for subschemes and are intimately related to moduli of sheaves and stable objects as in the theories developed by Mumford (Geometric Invariant Theory), Simpson, and Seshadri. Deformation-theoretic descriptions of tangent and obstruction spaces use Ext groups and the algebraic framework of Grothendieck duality and the cotangent complex developed by Quillen and Illusie, connecting to obstruction theories employed in virtual fundamental class constructions by Behrend and Fantechi. Relations with moduli of stable maps central to work by Kontsevich and Manin occur through comparisons with the Kontsevich moduli space and via gluing techniques used in degeneration arguments by Jun Li.

Applications in algebraic geometry and enumerative geometry

Hilbert schemes are tools for classical enumerative problems dating back to Schubert calculus and modern curve-counting theories like Gromov–Witten theory and Donaldson–Thomas theory pioneered by Kontsevich, Thomas, and Pandharipande. They underpin constructions of tautological classes used by Faber and computations in the cohomology rings studied by Lehn and Nakajima, the latter providing actions of infinite-dimensional algebras related to work of Heisenberg and representation-theoretic viewpoints of Grojnowski. Applications extend to birational geometry and derived categories as in research by Bondal, Orlov, and Bridgeland, and to string-theoretic enumerations connected with Mirror symmetry investigations by Candelas and Strominger.

Category:Algebraic geometry