The rank of the second Gaussian map for general curves

Abstract

Let X be a smooth, projective curve of genus g and let L be a line bundle on X. Consider the product X ×X, with the projections p1, p2 to the factors, and the natural morphism p to the symmetric product X(2). One has p∗(p ∗ 1L ⊗ p2L) = L ⊕ L−, where L± are the invariant and anti-invariant line bundles with respect to the involution (x, y) 7→ (y, x). One has H(L) ∼= SymH0(L) and H0(L−) ∼= ∧H(L). Restriction to the diagonal of X(2) gives rise to two maps