By Piotr Pragacz

The articles during this quantity are dedicated to:

- moduli of coherent sheaves;

- valuable bundles and sheaves and their moduli;

- new insights into Geometric Invariant Theory;

- stacks of shtukas and their compactifications;

- algebraic cycles vs. commutative algebra;

- Thom polynomials of singularities;

- 0 schemes of sections of vector bundles.

The major objective is to provide "friendly" introductions to the above issues via a chain of finished texts ranging from a really hassle-free point and finishing with a dialogue of present examine. In those texts, the reader will locate classical effects and strategies in addition to new ones. The e-book is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity idea. lots of the fabric offered within the quantity has no longer seemed in books before.

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Coherent sheaves on singular nonreduced curves and their moduli spaces have been studied (cf. -A. Inaba on moduli spaces of stable sheaves on reduced varieties of any dimension. In the case of curves we may hope of course much more detailed results. 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.

This follows immediately from the ﬁrst canonical ﬁltration. 3. We have R(M ) = lim p→∞ 1 dimC (M ⊗On,P On,P /(xp )) . 4. 1. Let 0 −→ M −→ M −→ M −→ 0 be an exact sequence of On,P -modules of ﬁnite type. Then we have R(M ) = R(M ) + R(M ). 2. Let 0 −→ E −→ E −→ E −→ 0 be an exact sequence of coherent sheaves on Cn . Then we have R(E) = R(E )+R(E ) and Deg(E) = Deg(E )+Deg(E ). Proof. 3. The assertion on degrees follows from the one on ranks and from the Riemann–Roch theorem. The generalized rank and degree are invariant by deformation.

If there is a universal sheaf on the whole M × X. 2. Endomorphisms Let S be an open set of sheaves on X admitting a ﬁne moduli space M . 1. For every E ∈ S, Aut(E) acts trivially on Ext1 (E, E) and dim(End(E)) is independent of E. Proof. For simplicity we assume that there is a globally deﬁned universal sheaf E on M × X. Using locally free resolutions we ﬁnd that there exists a morphism of vector bundles φ : A → B on M such that for every y ∈ M there is a canonical isomorphism End(Ey ) ker(φy ).