Let be an algebraically closed field, and a projective variety over . In the previous two posts, we’ve defined the Picard scheme , stated (without proof) the theorem of Grothendieck giving conditions under which it exists, and discussed the infinitesimal structure of (or equivalently of the connected component at the origin).

We saw in particular that the tangent space to the Picard scheme could be computed via

by studying deformations of a line bundle over the dual numbers. In particular, in characteristic zero, a simply connected smooth variety has trivial . To get interesting ‘s, we should be looking for non-simply connected varieties: abelian varieties are a natural example.

Let be an abelian variety over . The goal in this post is to describe , which we’ll call the **dual abelian variety** (we’ll see that it is in fact smooth). We’ll in particular identify the line bundles that it parametrizes. Most of this material is from David Mumford’s *Abelian varieties *and Alexander Polischuk’s *Abelian varieties, theta functions, and the Fourier transform. *I also learned some of it from a class that Xinwen Zhu taught last spring; my (fairly incomplete) notes from that class are here. (more…)