Derived Hilbert schemes

Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail

doi:10.1090/S0894-0347-02-00399-5

Derived Hilbert schemes
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by Ionuţ Ciocan-Fontanine and Mikhail M. Kapranov;

J. Amer. Math. Soc. 15 (2002), 787-815

DOI: https://doi.org/10.1090/S0894-0347-02-00399-5

Published electronically: June 21, 2002

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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 given by truncations of the homogeneous coordinate rings of subschemes in $X$. In particular, $RHilb_h(X)$ differs from $RQuot_n({\mathcal O_X})$, the derived Quot scheme constructed in our previous paper, which carries only a family of $A_\infty$-modules over the coordinate algebra of $X$. As an application, we construct the derived version of the moduli stack of stable maps of algebraic curves to a given projective variety $Y$, thus realizing the original suggestion of M. Kontsevich.

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