Coherent sheaves on multiple curves.

Jean-Marc Drézet

Collectanea Mathematica (2006)

  • Volume: 57, Issue: 2, page 121-171
  • ISSN: 0010-0757


This paper is devoted to the study of coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration C ⊂ C2 ⊂ ... ⊂ Cn = Y such that C is the reduced curve associated to Y, and for very P ∈ C there exists z ∈ OY,P such that (zi) is the ideal of Ci in OY,P. We define, using canonical filtrations, new invariants of coherent sheaves on Y: the generalized rank and degree, and use them to state a Riemann-Roch theorem for sheaves on Y. We define quasi locally free sheaves, which are locally isomorphic to direct sums of OCi, and prove that every coherent sheaf on Y is quasi locally free on some nonempty open subset of Y. We give also a simple criterion of quasi locally freeness. We study the ideal sheaves In,Z in Y of finite subschemes Z of C. When Y is embedded in a smooth surface we deduce some results on deformations of In,Z (as sheaves on S). When n = 2, i.e. when Y is a double curve, we can completely describe the torsion free sheaves on Y. In particular we show that these sheaves are reflexive. The torsion free sheaves of generalized rank 2 on C2 are of the form I2,Z ⊗ L, where Z is a finite subscheme of C and L is a line bundle on Y. We begin the study of moduli spaces of stable sheaves on a double curve, of generalized rank 3 and generalized degree d. These moduli spaces have many components. Sometimes one of them is a multiple structure on the moduli space of stable vector bundles on C of rank 3 and degree d.

How to cite


Drézet, Jean-Marc. "Faisceaux cohérents sur les courbes multiples.." Collectanea Mathematica 57.2 (2006): 121-171. <>.

author = {Drézet, Jean-Marc},
journal = {Collectanea Mathematica},
keywords = {Curvas algebraicas; Espacio de moduli; Haces vectoriales; non-reduced curves; coherent sheaves; moduli},
language = {fre},
number = {2},
pages = {121-171},
title = {Faisceaux cohérents sur les courbes multiples.},
url = {},
volume = {57},
year = {2006},

AU - Drézet, Jean-Marc
TI - Faisceaux cohérents sur les courbes multiples.
JO - Collectanea Mathematica
PY - 2006
VL - 57
IS - 2
SP - 121
EP - 171
LA - fre
KW - Curvas algebraicas; Espacio de moduli; Haces vectoriales; non-reduced curves; coherent sheaves; moduli
UR -
ER -

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