A Riemann singularities theorem for Prym theta divisors, with applications

Let (P,Xi) be the naturally polarized model of the Prym variety associated to the etale double cover pi : (C) over tilde -->C of smooth connected curv es, where Xi subset of P subset of Pic(2g-2) ((C) over tilde), and g (C) = g. If L is any "nonexceptional" singularity of Xi, i.e., a point L on Xi su bset of Pic(2g-2) ((C) over tilde) such that h(0)((C) over tilde, L) greate r than or equal to 4, but which cannot be expressed as pi* (M) (B) for any line bundle M on C with h(0)(C, M)greater than or equal to 2 and effective divisor B greater than or equal to0 on (C) over tilde, then we prove mult(L ) (Xi) = (1/2)h(0)((C) over tilde, L). We deduce that if C is nontetragonal of genus g greater than or equal to 11, then double points are dense in si ng(st) Xi= {L in Xi subset of Pic(2g-2) ((C) over tilde) such that h(0)((C) over tilde, L) greater than or equal to 4}. Let X = <(<alpha>)over tilde>( -1) (P) subset of Nm(-1) (|omega (C)|) where Nm : (C) over tilde ((2g-2) -- >) C(2g-2) is the norm map on divisors induced by pi, and <(<alpha>)over ti lde>: (C) over tilde ((2g-2))--> Pic(2g-2) ((C) over tilde) is the Abel map for (C) over tilde. If h : X --> |omega (C)| is the restriction of Nm to X and phi: X --> Xi is the restriction of <(<alpha>)over tilde> to X, and if dim(sing Xi) less than or equal to g-6, we identify the bundle h*(O(1)) de fined by the norm map h, as the line bundle T-phi circle times phi*(K-Xi) i ntrinsic to X, where T-phi is the bundle of tangents along the fibers of ph i. Finally we give a proof of the Torelli theorem for cubic threefolds, usi ng the Abel parametrization phi: X --> Xi.