  We prove that a very general hypersurface of bidegree (2, n) in P^2 x P^2 for
n bigger than or equal to 2 is not stably rational, using Voisin's method of
integral Chow-theoretic decompositions of the diagonal and their preservation
under mild degenerations. At the same time, we also analyse possible ways to
degenerate Prym curves, and the way how various loci inside the moduli space of
stable Prym curves are nested. No deformation theory of stacks or sheaves of
Azumaya algebras like in recent work of Hasset-Kresch-Tschinkel is used, rather
we employ a more elementary and explicit approach via Koszul complexes, which
is enough to treat this special case.
