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, 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 of two quadrics and in projective (2n+1)space, the variety of projective nspaces passing through turns out (over the complex numbers) to be an abelian variety. For this is pretty familiar, but, for general , 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 4descent.

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Dear Persiflage,
I cannot resist mentioning my favorite example of this kind of construction, which relates to smooth hyperplane sections H of G(4,8). H has dimension 15; it has primitive cohomology only in the middle degree, which gives a Hodge structure of dimension 6 and level 1. Thus the intermediate Jacobian is a PPAV.
Over the complex numbers, the PPAV which arise this way are exactly the Jacobians of the nonhyperelliptic curves X of genus 3. What about over a general field K of characteristic zero? Then the Jacobian of X arises from a Krational hyperplane section exactly when the curve X has a Krational flex in the canonical embedding.
The variety of P^n1’s in X is indeed an intermediate Jacobian – see e.g. this nice paper: http://archive.numdam.org/article/ASNSP_1980_4_7_2_217_0.pdf