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**Additional resources for Algebraic Cycles, Sheaves, Shtukas, and Moduli: Impanga Lecture Notes**

**Example text**

2. If the sheaves in S are simple then there exists a universal sheaf F on M × X. For every family E of sheaves of S parametrized by T there exists a line bundle L on T such that fE (F ) ⊗ p∗T (L) E. Proof. Suppose for simplicity that the sheaves in S are locally free. Then it is easy to build a universal projective bundle by gluing the projective bundles of all the local universal bundles, using the fact that the bundles of S are simple. Now this projective bundle comes from a universal bundle on M × X because a projective bundle which is banal on a nonempty open subset of the base is banal (cf.

The results of this paper come mainly from [7]. We introduce new invariants for coherent sheaves on multiple curves: the canonical ﬁltrations, generalized rank and degree, and prove a Riemann-Roch theorem. We deﬁne the quasi locally free sheaves which play the same role as locally free sheaves on smooth varieties. We study more precisely the coherent sheaves on double curves. In this case we can describe completely the torsion free sheaves of generalized rank 2, and give examples of moduli spaces of stable sheaves of generalized rank 3.

Vol. 29, N. 238 (1981). [21] Yoshioka, K. A note on the universal family of moduli of stable sheaves. Journ. Reine Angew. Math. 496 (1998), 149–161. fr/~drezet Algebraic Cycles, Sheaves, Shtukas, and Moduli Trends in Mathematics, 33–43 c 2007 Birkh¨ auser Verlag Basel/Switzerland Moduli Spaces of Coherent Sheaves on Multiples Curves Jean-Marc Dr´ezet 1. Introduction Let S be a projective smooth irreducible surface over C. The subject of this paper is the study of coherent sheaves on multiple curves embedded in S.