Pencils of quadrics and Jacobians of hyperelliptic curves

Using pencils of quadrics, we study a construction of torsors of Jacobians of hyperelliptic curves twice of which is Pic 1. We then use this construction to study the arithmetic invariant theory of the actions of SO2n+1 and PSO2n+2 on self-adjoint operators and show how they facilitate in computing the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves with a rational Weierstrass point, and the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves with a rational non-Weierstrass point, over arbitrary number fields.