If Σ is a smooth genus two
curve, Σ ⊂Pic1(Σ) the Abel embedding in the degree one Picard variety, |2Σ| the
projective space parametrizing divisors on Pic1(Σ) linearly equivalent to
2Σ, and Pic0(Σ)2= G≅(ℤ∕2ℤ)4 the subgroup of points of order two in the
Jacobian variety J(Σ) =Pic0(Σ), then G acts on |2Σ| and the quotient variety
|2Σ|∕G parametrizes two fundamental moduli spaces associated with the
curve Σ. Namely, Narasimhan-Ramanan’s work implies an isomorphism of
|2Σ|∕G with the space ℳ of (S-equivalence classes of semi-stable, even) ℙ1 bundles over Σ,
and Verra has defined a precise birational correspondence between |2Σ|∕G and Beauville’s
compactification of 𝒫−1(J(Σ)) the fiber of the classical Prym map over
J(Σ). In this paper we give a new (birational) construction of the composed
Narasimhan-Ramanan-Verra map α : ℳ−−→𝒫−1(J(Σ)), defined purely in terms of
the geometry of a (generic stable) ℙ1 bundle X → Σ in ℳ, and also an explicit
rational inverse map β : 𝒫−1(J(Σ))−−→ℳ. The map α may be viewed as an analog
for Prym varieties of Andreotti’s reconstruction of a curve C of genus g from the
branch locus of the canonical map on the symmetric product C(g−1). The map β
assigns to an étale double cover π :C→ C in 𝒫−1(J(Σ)), where C and C are curves
of genera 5 and 3 respectively, the ℙ1 bundle φ : X → Σ, where X = {divisors D in
C(4): π∗(D) ≡ ωC, and h0(D) is even} and φ : X → φ(X)≅Σ ⊂Pic4(C) is the Abel
map.
