By Piotr Pragacz

The articles during this quantity are committed to:

- moduli of coherent sheaves;

- crucial 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 offer "friendly" introductions to the above subject matters via a sequence of entire texts ranging from a really undemanding 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 booklet is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity thought. lots of the fabric awarded within the quantity has no longer seemed in books before.

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

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

**Example text**

M . It follows that a coherent sheaf E of positive rank is semi-stable (resp. , it has no subsheaf with ﬁnite support) and if for every proper subsheaf F ⊂ E we have Deg(E) Deg(F ) ≤ R(F ) R(E) (resp. < ). Moduli Spaces of Coherent Sheaves on Multiples Curves 37 4. Quasi locally free sheaves Let P ∈ C and z ∈ On,P be a local equation of C. Let M be a On,P -module of ﬁnite type. Then M is called quasi free if there exist nonnegative integers m1 , . . , mn and an isomorphism M ⊕ni=1 mi Oi,P . The integers m1 , .

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 ). Now let p = rk(ker(φ)), it is the generic dimension of Aut(E), for E ∈ S. Now let E ∈ S and y ∈ M such that Ey E. Let W be the image of ker(φ)y in Ay . Then we have dim(W ) ≤ p. Let F be a family of sheaves in S parametrized by an algebraic variety, and s ∈ S such that Fs E.

Some object J which provides a bijection between morphisms to it and families of line bundles up to isomorphism (not up to equivalence). The answer is yes, but this object J is not a scheme! It is an algebraic stack (in the sense of Artin): the Jacobian stack. A stack is a generalization of the notion of scheme, but we will not consider it here. We would like to have a scheme with the same properties as the Jacobian, but for torsion-free sheaves instead of line bundles. To be able to do this, we have to consider only the semistable ones.