On the Singularities of Theta Divisors on Jacobians

The theta divisor Θ of the Jacobian variety of a complex curve X is best viewed as a divisor inside the component Pic(X) consisting of (isomorphism classes of) line bundles of degree g−1. Then a line bundle L belongs to Θ if and only if it has non-zero sections. Riemann proved that the multiplicity of Θ at a point L is equal to dimH(X;L) − 1. Kempf ([Ke]) obtained a geometric proof of Riemann’s theorem and a beautiful description of the tangent cone to Θ at any point. In this paper we study the intersection cohomology of Θ when X is not hyperelliptic. Our starting point is a theorem of Martens concerning the geometry of the Abel-Jacobi mapping φ : S(X) → Pic(X) and of its fibers. We interpret this theorem as saying that φ is small in the sense of Goresky and MacPherson. This means that the intersection cohomology IH(Θ;Q) is isomorphic to the cohomology H(S(X);Q). The cohomology of S(X), including the algebra structure, was completely determined by MacDonald in [Mac] . From the evaluation of the differential of φ we deduce (Theorem 3.3.1) that the intersection complex has the property that its characteristic variety (inside the cotangent bundle of Pic(X)) is irreducible. This is a rather unusual phenomenon; it is known to be true for Schubert varieties in classical grassmannians ([BFL]) and more generally in hermitian symmetric spaces of simply-laced groups ([BF]). In Section 4 we study the effect on the intersection cohomology of the involution ι of Θ given by