Brill-Noether theory of curves with A(k) singularities and hyperplane sections of K3 surfaces

Let Y be an integral projective curve whose singularities are of type A(k), i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= p (a)(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned d egree d line bundles, L, with h(0)(Y, L) greater than or equal to r + 1 eve n if the Brill - Noether number rho (g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane s ection of a simple K3 surface S subset of P-g.