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Derived Hilbert schemes


Authors: Ionut Ciocan-Fontanine and Mikhail M. Kapranov
Journal: J. Amer. Math. Soc. 15 (2002), 787-815
MSC (2000): Primary 14M30; Secondary 18G50
DOI: https://doi.org/10.1090/S0894-0347-02-00399-5
Published electronically: June 21, 2002
MathSciNet review: 1915819
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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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Additional Information

Ionut Ciocan-Fontanine
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: ciocan@math.umn.edu

Mikhail M. Kapranov
Affiliation: Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario, Canada M5S 3G3
Email: kapranov@math.toronto.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00399-5
Received by editor(s): August 14, 2000
Published electronically: June 21, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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