Derived Hilbert schemes

Abstract

We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X X with given Hilbert polynomial h h . This is a dg-manifold (smooth dg-scheme) R H i l b h ( X ) RHilb_h(X) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is given by truncations of the homogeneous coordinate rings of subschemes in X X . In particular, R H i l b h ( X ) RHilb_h(X) differs from R Q u o t n ( O X ) RQuot_n({\mathcal O_X}) , the derived Quot scheme constructed in our previous paper, which carries only a family of A ∞ A_\infty -modules over the coordinate algebra of X X . As an application, we construct the derived version of the moduli stack of stable maps of algebraic curves to a given projective variety Y Y , thus realizing the original suggestion of M. Kontsevich.