By Piotr Pragacz

The articles during this quantity are dedicated 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 function is to provide "friendly" introductions to the above themes via a sequence of entire texts ranging from a truly effortless point and finishing with a dialogue of present study. 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 idea. lots of the fabric provided within the quantity has now not seemed in books before.

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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 ). If σ ∈ Ext1 (E, E ) let 0 −→ E −→ Fσ −→ E −→ 0 (∗) be the corresponding extension. The group G = Aut(E)× Aut(E ) acts obviously on Ext1 (E, E ) and if σ ∈ Ext1 (E, E ) and g ∈ G we have Fgσ = Fσ . Let σ ∈ Ext1 (E, E ). The tangent map at the identity of the orbit map Φσ : G g / Ext1 (E, E ) / gσ is TΦσ : End(E) × End(E )  (α, β) / Ext1 (E, E ) / βσ − σα. We say that (∗) is a generic extension if TΦσ is surjective.

Then the point Cφ of P(W ) is called semi-stable (resp. stable) with respect to t if – im(φ) is not contained in O(−1) ⊗ C7 , – For every proper linear subspace D ⊂ C7 , im(φ) is not contained in O(−2) ⊕ (O(−1) ⊗ D). – For every 1-dimensional linear subspace L ⊂ C2 , if K ⊂ C, D ⊂ C7 are linear subspaces such that φ(O(−3) ⊗ L) ⊂ (O(−2) ⊗ K) ⊕ (O(−1) ⊗ D), then we have 1 1−t dim(D) ≥ t dim(K) + 7 2 (resp. >). Let P(W )ss (t) (resp. P(W )s (t)) denote the open set of semi-stable (resp. stable) points of P(W ) with respect to t.

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. stable) then γ − 2l ≤ ≤ l + γ (resp.

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