The Gaussian map for rational ruled surfaces

Abstract

In this paper the Gaussian map Φ : ∧ 2 H 0 ( C , K ) → H 0 ( C , 3 K ) \Phi :{ \wedge ^2}{H^0}(C,K) \to {H^0}(C,3K) of a smooth curve C C lying on a minimal rational ruled surface is computed. It is shown that the corank of Φ \Phi is determined for almost all such curves by the rational surface in which it lies. Hence, except for some special cases, a curve cannot lie on two nonisomorphic minimal rational ruled surfaces.