Derived Hilbert schemes

2000/05/16 by I. Ciocan-Fontanine, Ionuţ Ciocan-Fontanine, Mikhail Kapranov +3 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.math/0005155

37 pages, AMS-Tex

arxiv created 2000/05/16 · openalex publication_date 2000/05/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Abel maps for integral curves via a derived perspective

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