Abelian Varieties

Jerry Wang gave a nice talk this week on his generalization of Manjul’s work on pointless hyperelliptic curves to hyperelliptic curves with no points over any field of odd degree (equivalently, \(\mathrm{Pic}^1\) is pointless). This work (link here) is joint with Manjul and Dick, so the exposition is predictably of high quality. But I wanted to mention a result that arose during the talk which I found quite intriguing. Namely, given the intersection \(X\) of two quadrics \(P\) and \(Q\) in projective (2n+1)-space, the variety of projective n-spaces passing through \(X\) turns out (over the complex numbers) to be an abelian variety. For \(n = 1\) this is pretty familiar, but, for general \(n\), I hadn’t seen any construction like this before. It gives, for example, explicit constructions of equations for abelian varieties in surprisingly low degree. It brought me back to a lecture I once went to by Beauville as a graduate student when he talked about intermediate Jacobians (wait – perhaps this construction also has to be isomorphic to an intermediate Jacobian…). Is it possible (in some weak sense) to classify all varieties whose variety of maximal linear subspaces is an abelian variety of suitably high dimension? Are there varieties in which this construction gives rise to abelian varieties which are not isogenous to Jacobians? The geometric result is due (independently) to several authors, but, in a solo paper here, Jerry showed that the result is true arithmetically, and, even better, the construction can more precisely be described as giving an explicit torsor for the corresponding Jacobian. This very nicely generalizes the classical picture between pairs of quadrics and 2- and 4-descent.