A Riemann Singularities Theorem for Prym Theta Divisors, with Applications

Let (P,Ξ) be the naturally polarized model of the Prym variety associated to the étale double cover π : C̃ → C of smooth connected curves, where Ξ ⊂ P ⊂ Pic2g−2(C̃), and g(C) = g. If L is any “non exceptional” singularity of Ξ, i.e. a point L on Ξ ⊂ Pic2g−2(C̃) such that h0(C̃, L) ≥ 4, but which cannot be expressed as π∗(M)(B) for any line bundle M on C with h0(C,M) ≥ 2 and effective divisor B ≥ 0 on C̃, then we prove multL(Ξ) = (1/2)h0(C̃, L). We deduce that if C is non tetragonal of genus g ≥ 11, then double points are dense in singstΞ = {L in Ξ ⊂ Pic2g−2(C̃) such that h0(C̃, L) ≥ 4}. Let X = α̃−1(P ) ⊂ Nm−1(|ωC |) where Nm : C̃(2g−2) → C(2g−2) is the norm map on divisors induced by π, and α̃ : C̃(2g−2) → Pic2g−2(C̃) is the Abel map for C̃. If h : X → |ωC | is the restriction of Nm to X and φ : X → Ξ is the restriction of α̃ to X, and if dim(singΞ) ≤ g − 6, we identify the bundle h∗(O(1)) defined by the norm map h, as the line bundle Tφ ⊗ φ∗(KΞ) intrinsic to X, where Tφ is the bundle of “tangents along the fibers” of φ. Finally we give a proof of the Torelli theorem for cubic threefolds, using the Abel parametrization φ : X → Ξ.