By Piotr Pragacz

ISBN-10: 3764385367

ISBN-13: 9783764385361

ISBN-10: 3764385375

ISBN-13: 9783764385378

The articles during this quantity are dedicated to:

- moduli of coherent sheaves;

- critical 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 goal is to provide "friendly" introductions to the above themes via a chain of complete texts ranging from a really ordinary point and finishing with a dialogue of present examine. In those texts, the reader will locate classical effects and techniques in addition to new ones. The ebook is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity conception. many of the fabric provided within the quantity has no longer seemed in books before.

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

**Example text**

7. Local structure of torsion free sheaves. Let M be a torsion free O2,P module. Then there exist integers m, q, n1 , . . , np such that p Ini ,P M ⊕ mO2,P ⊕ qOC,P . 3. Deformations of sheaves If E is a coherent sheaf on C then the canonical morphism Ext1On (E, E) −→ Ext1OS (E, E) is an isomorphism. Let M be a O2,P -module, M2 ⊂ M its canonical ﬁltration. Let r0 (M ) = rk(M2 ). Then we have R(M ) ≥ 2r0 (M ). If M is quasi free then we have R(M ) = 2r0 (M ) if and only if M is free. 1. Let M be a quasi free O2,P -module, and r0 an integer such that 0 < 2r0 ≤ R(M ).

Let Y ⊂ X × X be the set of pairs (E, E ) such that Ext1 (E , E) = Ext2 (E , E ) = Ext2 (E, E) = Ext2 (E , E) = Exti (E, E ) = {0} for i ≥ 2 and that there exists a generic extension of E by E . Let Y the set of such generic extensions. Then it is easy to see that the map Y → Y associating to (E, E ) the generic extension Fσ of E by E is a bijection. 1. The set Y is open and admits a ﬁne moduli space N which is isomorphic to the open subset of M × M consisting of pairs (s, t) such that (Es , Et ) ∈ Y .

We have rk(EE ) = rk(ΓE ) = 1 and rk(FE ) = rk(GE ) = 2. The ranks and degrees of EE , FE , GE and ΓE are invariant by deformation of E. 1. The sheaf E is (semi-)stable if and only if (i) For every sub-line bundle D of GE we have deg(D ) ≤ μ(E) (resp. <). (ii) For every quotient line bundle D of FE we have μ(E) ≤ deg(D ) (resp. <). It follows that if FE and GE are stable then so is E. Let = deg(EE ), γ = deg(ΓE ). We have then Deg(E) = 2 + γ + l. By considering the subsheaves EE , GE of E we ﬁnd that if E is semi-stable (resp.

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

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