Tuesday, November 14, 2017
4:00 pm

Mathematics Colloquium

Autoduality of Jacobians for singular curves
Dima Arinkin, Department of Mathematics, University of Wisconsin-Madison

Let C be a (smooth projective algebraic) curve, that is, a Riemann surface. The Jacobian J of C is a natural geometric object attached to C: it parametrizes line bundles on C. It is a classical result that J is self-dual: it is identified with a space of line bundles on itself.

Suppose now that C is singular. We can still consider the Jacobian J of C, but it turns out to be no longer compact, which makes self-duality impossible. However, there is a natural way to compactify J by considering torsion-free sheaves instead of line bundles. The result is the compactified Jacobian, which I will denote J'.

In this talk, I consider curves C with planar singularities. The main result is that J' is still self-dual: J' is identified with a space of torsion-free sheaves on itself. This autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk.

Contact Mathematics Department mathinfo@caltech.edu at 626-395-4335
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