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Abelian Varieties learning seminar

Thursday, July 11, 2013 — 2:30 PM EDT

Robert Garbary, Pure Mathematics University of Waterloo

“Quotient Spaces”

Suppose we have an algebraic group G acting algebraicly on a variety X, ie for each g ∈ G the associated map X → X is a morphism. A quotient of X by G is defined to be a variety Y and a morphism π : X → Y satisfying
(1) π−1(π(x)) = Gx for all x ∈ X.
(2) For any variety Z and G-invariant morphism X → Z, there is a unique factorization through Y.
Quotients do not always exist - for example, consider C∗ acting on C (or more generally, any toric variety). One necessary but insufficient condition is that all orbits be closed.
I’m going to prove that if G is finite then the quotient always does exist. The main ingredient is a classical result of Hilbert from Invariant Theory. In particular, this proves that the set SymN(X) is actually a variety, which is the beginning of a proof that Jac(C) is a variety.

Location
MC - Mathematics & Computer Building
5046
200 University Avenue West

Waterloo, ON N2L 3G1

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