A characterization of non-hyperelliptic Jacobi varieties.
Let be a compact complex nonsingular surface without curves, and a holomorphic vector bundle of rank 2 on . It turns out that the associated projective bundle has no divisors if and only if is “strongly” irreducible. Using the results concerning irreducible bundles of [Banica-Le Potier, J. Crelle, 378 (1987), 1-31] and [Elencwajg- Forster, Annales Inst. Fourier, 32-4 (1982), 25-51] we give a proof of existence for bundles which are strongly irreducible.
A version of the classical Nakai-Moishezon criterion is proved for all compact complex surfaces, regardless of the parity of the first Betti number.