Derived Hilbert schemes

TL;DR

This paper constructs a derived version of the Hilbert scheme as a smooth dg-scheme, enriching the classical structure with a natural family of commutative dg-algebras, and extends this to a derived moduli stack of stable maps.

Contribution

It introduces the derived Hilbert scheme as a dg-manifold with a natural family of commutative dg-algebras, and constructs the derived moduli stack of stable maps, extending classical moduli theory.

Findings

Constructed the derived Hilbert scheme as a smooth dg-scheme.

Established a natural family of commutative dg-algebras over the derived Hilbert scheme.

Built the derived moduli stack of stable maps of algebraic curves.

Abstract

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is just given by truncations of the homogeneous coordinate rings of subschemes in X. In particular, RHilb_h(X) differs from RQuot_h(O_X), the derived Quot scheme constructed in our previous paper (math.AG/9905174) which carries only a family of A-infinity modules over the coordinate algebra of X. As an application, we construct the derived version of the moduli stack of stable maps of (variable) algebraic curves to a given projective variety Y, thus realizing the original suggestion of M. Kontsevich.