THE COMPACTIFIED JACOBIAN OF A REDUCIBLE HYPERELLIPTIC CURVE

Abstract

There is a one-to-one correspondence between nonsingular pencils of quadrics in an odd-dimensional projective space and nonsingular hyperelliptic curves. There is also a close relation between the moduli spaces of vector bundles of ranks 1, 2 on the hyperelliptic curve and spaces associated to the intersection of quadrics of the corresponding pencil. This has been studied extensively by many authors [1, 5, 8, 9, 11]. An interesting application of this was in the proof of the Verlinde formulae by Szenes [12]. This relationship was generalised to a relation between spaces related to singular pencils of quadrics associated to irreducible nodal curves and moduli of torsion-free sheaves on the nodal curves [2, 4]. The projective models of the moduli of vector bundles on smooth curves (given by this relationship) led to explicit representations of integrable systems of [6, 7]. Similar results on singular curves would provide important classes of special solutions. The pencils of quadrics considered in previous works had associated Segre symbols of type [1 … 1] and [2 … 2 1 … 1]. Kleiman (and Emma Previato) had raised the question: What happens in the case of Segre symbol [2 2 2]? In this case the associated curve Y has two rational components intersecting transversely at three points. Unlike in the previous cases, the compactified Jacobian has more than one component. Moreover the stability of line bundles and the compactification of the Jacobian depends on the choice of a polarisation on Y, making the problem more interesting and difficult.