Singular pencils of quadrics and compactified Jacobians of curves

Abstract

Let Y be an irreducible nodal hyperelliptic curve of arithmetic genus g such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in P^2g+1 with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian of Y is isomorphic to the space R of (g-1) dimensional linear subspaces P^2g+1 of which are contained in the intersection Q of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian of Y is isomorphic to the open subset of R consisting of the (g-1) dimensional subspaces not passing through any singular point of Q.