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.
@article{urn:eudml:doc:41834,
title = {Faisceaux coh\'erents sur les courbes multiples.},
journal = {Collectanea Mathematica},
volume = {57},
year = {2006},
pages = {121-171},
zbl = {1106.14019},
mrnumber = {MR2223850},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41834}
}
Drézet, Jean-Marc. Faisceaux cohérents sur les courbes multiples.. Collectanea Mathematica, Tome 57 (2006) pp. 121-171. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41834/