2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132202Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our study of Brill-Noether theory of curves on Enriques surfaces (reference [KL1]) and of Enriques-Fano threefolds (reference [KLM]).4 pages, latex. Minor corrections. To appear on Proc. Amer. Math. SocAlgebraic Geometry14F17, 14J28, 14C20text