Algebraic Cycles, Sheaves, Shtukas, and Moduli: Impanga by Piotr Pragacz

By Piotr Pragacz

Articles research the contributions of the nice mathematician J. M. Hoene-Wronski. even if a lot of his paintings used to be brushed aside in the course of his lifetime, it's now well-known that his paintings deals helpful perception into the character of arithmetic. The publication starts with elementary-level discussions and ends with discussions of present examine. lots of the fabric hasn't ever been released ahead of, supplying clean views on Hoene-Wronski’s contributions.

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Clearly, a geometric quotient is a good quotient, and a good quotient is a categorical quotient. Geometric Invariant Theory (GIT) is a technique to construct good quotients (cf. [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.

From this it follows that slope-stable =⇒ stable =⇒ semistable =⇒ slope-semistable Note that, if n = 1, Gieseker and Mumford (semi)stability coincide, because the Hilbert polynomial has degree 1. Let E be a torsion free sheaf on X. There is a unique filtration, called the Harder-Narasimhan filtration, 0 = E0 E1 E2 ··· El = E Lectures on Principal Bundles over Projective Varieties 47 such that E i = Ei /Ei−1 is semistable and P i+1 PE i > E i+1 i rk E rk E for all i. In particular, any torsion free sheaf can be described as successive extensions of semistable sheaves.

We call Fσ the generic extension of E by E . 1. Suppose that Hom(E , E) = Ext1 (E , E) = Ext2 (E, E ) = {0} and let σ ∈ Ext1 (E, E ) such that Fσ is a generic extension. Then we have Ext1 (Fσ , Fσ ) Ext1 (E , E ) ⊕ Ext1 (E, E) and an exact sequence 0 −→ Hom(E, E ) −→ End(Fσ ) −→ ker(Φσ ) −→ 0. Proof. This follows easily from a spectral sequence associated to the filtration 0 ⊂ E ⊂ Fσ . In other words, the only deformations of Fσ are generic extensions of deformations of E by deformations of E . 2.

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