By Piotr Pragacz

ISBN-10: 3764385367

ISBN-13: 9783764385361

ISBN-10: 3764385375

ISBN-13: 9783764385378

The articles during this quantity are committed to:

- moduli of coherent sheaves;

- vital 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 function is to offer "friendly" introductions to the above subject matters via a sequence of entire texts ranging from a really common point and finishing with a dialogue of present study. In those texts, the reader will locate classical effects and techniques in addition to new ones. The booklet is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity thought. lots of the fabric provided within the quantity has now not seemed in books before.

**Read or Download Algebraic cycles, sheaves, shtukas, and moduli PDF**

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**Additional info for Algebraic cycles, sheaves, shtukas, and moduli**

**Example text**

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. [12]). -M. Dr´ezet 3. 1. Generic extensions Let E , E be coherent sheaves on X. The extensions of E by E are parametrized by Ext1 (E, E ).

M. D´eformations des extensions larges de faisceaux. Paciﬁc Journ. of Math. 220, 2 (2005), 201–297. S. Groupe de Picard des vari´ et´es de modules de ﬁbr´es semi-stables sur les courbes alg´ebriques. Invent. Math. 97 (1989), 53–94. , Le Potier, J. Fibr´es stables et ﬁbr´es exceptionnels sur P2 . Ann. Ec. Norm. Sup. 18 (1985), 193–244. , Trautmann, G. Moduli spaces of decomposable morphisms of sheaves and quotients by nonreductive groups. Ann. de l’Inst. Fourier 53 (2003), 107–192. [9] Godement, R.

Let n be a positive integer. Let Z be a ﬁnite subscheme of P2 such that h0 (OZ ) = n(n+1) , and IZ be its ideal sheaf. Then it follows easily from the Beilin2 son spectral sequence that Z is not contained in a curve of degree n − 1 if and only if there is an exact sequence 0 −→ O(−n − 1) ⊗ Cn −→ O(−n) ⊗ Cn+1 −→ IZ −→ 0. Let W = Hom(O(−n−1)⊗Cn , O(−n)⊗Cn+1 ). We consider the action of the reductive group G = SL(n)×SL(n+1) on P(W ), with the obvious linearization. If f ∈ W \{0}, then Cf is semi-stable if and only if Cf is stable, if and only if for any two linear subspaces H ⊂ Cn , K ⊂ Cn+1 , of dimension p > 0, f (O(−n − 1) ⊗ H) Exotic Fine Moduli Spaces of Coherent Sheaves 29 is not contained in O(−n) ⊗ K.

### Algebraic cycles, sheaves, shtukas, and moduli by Piotr Pragacz

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