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 ⊂ Pic2g−2(C), and g(C) = g. If L is any “nonexceptional” singularity of
Ξ, i.e., a point L on Ξ ⊂ Pic2g−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 multL(Ξ) = (1∕2)h0(C,L). We deduce
that if C is nontetragonal of genus g ≥ 11, then double points are dense in
singstΞ = {L in Ξ ⊂ Pic2g−2(C) such that h0(C,L) ≥ 4}. Let
X =α−1(P) ⊂ Nm−1(|ωC|) where Nm:C(2g−2)→ C(2g−2) is the norm map on
divisors induced by π, and α:C(2g−2)→ Pic2g−2(C) is the Abel map for C. If
h:X →|ωC| 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 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.
