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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Derived Hilbert schemes
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by Ionuţ Ciocan-Fontanine and Mikhail M. Kapranov PDF
J. Amer. Math. Soc. 15 (2002), 787-815 Request permission

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
  • Ionuţ Ciocan-Fontanine
  • Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • MR Author ID: 365502
  • Email: ciocan@math.umn.edu
  • Mikhail M. Kapranov
  • Affiliation: Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario, Canada M5S 3G3
  • MR Author ID: 200368
  • Email: kapranov@math.toronto.edu
  • Received by editor(s): August 14, 2000
  • Published electronically: June 21, 2002
  • © Copyright 2002 American Mathematical Society
  • 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
  • MathSciNet review: 1915819