Geodesic
Derived Hilbert schemes
Ciocan-Fontanine, Ionuţ1 ; Kapranov, Mikhail2
1Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
2Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario, Canada M5S 3G3
Journal of the American Mathematical Society, Tome 15 (2002) no. 4, pp. 787-815
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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.
DOI : 10.1090/S0894-0347-02-00399-5

Ciocan-Fontanine, Ionuţ  1   ; Kapranov, Mikhail  2

1 Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
2 Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario, Canada M5S 3G3 
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Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail. Derived Hilbert schemes. Journal of the American Mathematical Society, Tome 15 (2002) no. 4, pp. 787-815. doi: 10.1090/S0894-0347-02-00399-5

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