Abstract

Let (P,Ξ) be the naturally polarized model of the Prym variety associated to the étale double cover π: C → C of smooth connected curves, where Ξ ⊂ P ⊂ Pic^2g−2(C), and g(C) = g. If L is any “nonexceptional” singularity of Ξ, i.e., a point L on Ξ ⊂ Pic^2g−2(C) such that h⁰(C,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 C, then we prove mult_L(Ξ) = (1 ∕ 2)h⁰(C,L). We deduce that if C is nontetragonal of genus g ≥ 11, then double points are dense in sing_stΞ = {L in Ξ ⊂ Pic^2g−2(C) such that h⁰(C,L) ≥ 4}. Let X = α⁻¹(P) ⊂ Nm⁻¹(|ω_C|) where Nm: C^(2g−2) → C^(2g−2) is the norm map on divisors induced by π, and α: C^(2g−2) → Pic^2g−2(C) is the Abel map for C. If h: X →|ω_C| is the restriction of Nm to X and ϕ: X → Ξ is the restriction of α 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 T_ϕ × ϕ^*(K_Ξ) intrinsic to X, where T_ϕ 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.

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