Faisceaux cohérents sur les courbes multiples

• Autores: Jean-Marc Drézet
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. 2, 2006, págs. 121-171
• Idioma: francés
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• Resumen

  • 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 every 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.