Mathematics Colloquium

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.