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;

- imperative 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 provide "friendly" introductions to the above subject matters via a chain of accomplished texts ranging from a truly straightforward 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 publication is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity thought. many of the fabric offered within the quantity has no longer seemed in books before.

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

**Sample text**

A categorical quotient is a scheme M with a G-invariant morphism p : R → M such that for every other scheme M , and G-invariant morphism p , there is a unique morphism ϕ with p = ϕ ◦ p RC CC CCp p CC C! 2 (Good quotient). Let R be a scheme endowed with a G-action. A good quotient is a scheme M with a G-invariant morphism p : R → M such that 1. p is surjective and aﬃne G G ) = OM , where OR is the sheaf of G-invariant functions on R. 2. p∗ (OR 3. If Z is a closed G-invariant subset of R, then p(Z) is closed in M .

3 (Moduli space). We say that M is a moduli space for a set of objects, if it corepresents the functor of families of those objects. 4 (Coarse moduli). A scheme M is called a coarse moduli scheme for F if it corepresents F and furthermore the map φ(Spec C) : F (Spec C) → Hom(Spec C, M ) is bijective. , the functor deﬁned as F (T ), if T = Spec C F (T ) = S-equivalence classes of objects of F (Spec C), if T = Spec C 1. Moduli space of torsion free sheaves In this section we will sketch the proof of the existence of the moduli space of semistable torsion free sheaves.

In Contemporary Mathematics 58, Proc. of Lefschetz Centennial Conf. (1984), AMS, 47–64. [2] Bhosle Usha N. Generalized parabolic bundles and applications to torsion free sheaves on nodal curves. Arkiv for Matematik 30 (1992), 187–215. [3] Bhosle Usha N. Picard groups of the moduli spaces of vector bundles. Math. Ann. 314 (1999) 245–263. , Ongay, F. Nonemptiness of BrillNoether loci. Intern. J. Math. 11 (2000), 737–760. -M. Faisceaux coh´erents sur les courbes multiples. Collectanea Mathematica 57, 2 (2006), 121–171.

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

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