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

By Piotr Pragacz

The articles during this quantity are dedicated to:

- moduli of coherent sheaves;

- important 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 themes via a sequence of accomplished texts ranging from a truly 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 ebook is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity idea. many of the fabric provided within the quantity has now not seemed in books before.

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

**Example text**

4. 1. Introduction Let OX (1) be a very ample line bundle on X. We consider extensions of sheaves 0 −→ G −→ E −→ F −→ (∗) where G, E are locally free and F torsionfree. We want to see what happens when μ(G) μ(E), say let G = G0 (p) and let p tend to inﬁnity. If we want nontrivial extensions F must not be locally free. The correct settings are as follows: F is torsionfree and T = F ∗∗ /F is pure of codimension 2. Using the dual of (∗) we obtain the exact sequence 0 / F∗ We have Ext1 (F, O) same support as T .

5. 1. Maximal ﬁne moduli spaces Let S be an open set of sheaves admitting a ﬁne moduli space M . Then S (or M ) is called maximal if there does not exist an open set of sheaves T , such that S ⊂ T , S = T , admitting a ﬁne moduli space. Of course if M is projective then S and M are maximal. We will give an example that shows that the converse is not true. 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.

3. 2 that if deg(L) < 0 we have an exact sequence of abelian groups 0 / H 1 (Ln−1 ) / Pic(Cn ) rn / Pic(Cn−1 ) / 0, where rn is the restriction morphism. Let Pn ⊂ Pic(Cn ) be the subgroup consisting of line bundles whose restriction to C is OC . Then we have a ﬁltration of abelian groups O = G0 ⊂ G1 ⊂ · · · ⊂ Gn−1 = Pn such that Gi /Gi−1 H 1 (Li ) for 1 ≤ i ≤ n − 1. Here Gi is the subgroup of Pn of line bundles whose restriction to Cn−i is trivial. , to a ﬁnite-dimensional vector space. Moduli Spaces of Coherent Sheaves on Multiples Curves 35 3.