Derived Hilbert schemes
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.
Cite
@article{arxiv.math/0005155,
title = {Derived Hilbert schemes},
author = {I. Ciocan-Fontanine and M. Kapranov},
journal= {arXiv preprint arXiv:math/0005155},
year = {2007}
}
Comments
37 pages, AMS-Tex