Benedict H. Gross
Harvard University

Title: On the arithmetic of pencils of quadrics
Abstract: The Fano variety of maximal linear subspaces of the complete intersection
of two quadrics in projective space of dimension 2n+1 is a principal homogenous
space for the Jacobian of a hyperelliptic curve of genus n. I will review this theory
over the complex numbers, then make it more precise for quadrics over an arbitrary field, 
with characteristic is not equal to 2. I will also discuss an arithmetic application of
this theory. This relates certain rational orbits in a representation of the split orthogonal group
SO(2n+1) and classes in the 2-Selmer groups of hyperelliptic curves of genus n over Q with
a rational Weierstrass point.