By Piotr Pragacz

The articles during this quantity are committed to:

- moduli of coherent sheaves;

- relevant 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 offer "friendly" introductions to the above themes via a chain of entire texts ranging from a truly user-friendly point and finishing with a dialogue of present study. In those texts, the reader will locate classical effects and techniques in addition to new ones. The e-book is addressed to researchers and graduate scholars in algebraic geometry, algebraic topology and singularity conception. many of the fabric offered within the quantity has no longer seemed in books before.

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

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

**Sample text**

3. Canonical ﬁltrations are preserved by morphisms of sheaves. 4. Examples. 1. If E is locally free and E = E|C , then Ei = E (i) and Ei /Ei+1 = E ⊗ Li−1 for 1 ≤ i ≤ n. 2. If E is the ideal sheaf of a ﬁnite subscheme T of C then Ei /Ei+1 = (OC (−T )⊗ Li−1 ) ⊕ OT if 1 ≤ i < n, En = OC (−T ) ⊗ Ln−1 , E (i) /E (i+1) = Li−1 if 2 ≤ i ≤ n and E (1) /E (2) = OC (−T ). 2. Generalized rank and degree and Riemann–Roch theorem The integer R(M ) = rk(Gr(M )) is called the generalized rank of M . The integer R(E) = rk(Gr(E)) is called the generalized rank of E, and Deg(E) = deg(Gr(E)) is called the generalized degree of E.

We will construct another smooth projective ﬁne moduli space M which has a nonempty intersection with M (6, −3, 8). Stable sheaves of rank 6 and Chern classes −3, 8 are related to morphisms O(−3) ⊗ C2 −→ O(−2) ⊕ (O(−1) ⊗ C7 ). Let W be the vector space of such morphisms. We consider the action of the nonreductive group G = Aut(O(−3) ⊗ C2 ) × Aut(O(−2) ⊕ (O(−1) ⊗ C7 )) on P(W ). Let H = (I, I 0 ) ; α I α ∈ Hom(O(−2), O(−1) ⊗ C7 ) be the unipotent subgroup of G, and Gred = Aut(O(−3) ⊗ C2 ) × Aut(O(−2)) × Aut(O(−1) ⊗ C7 ), which is a reductive subgroup of G.

Mu1]). Assume R is projective, and the action of G on R has a linearization on an ample line bundle OR (1). A closed point z ∈ R is called GIT-semistable if, for some m > 0, there is a G-invariant section s of OR (m) such that s(z) = 0. If, moreover, the orbit of z is closed in the open set of all GIT-semistable points, it is called GIT-polystable, and, if furthermore, this closed orbit has the same dimension 52 T. , if z has ﬁnite stabilizer), then z is called a GIT-stable point. We say that a closed point of R is GIT-unstable if it is not GIT-semistable.