On secant spaces to Enriques surfaces

Abstract

We relate the minimal gonality of smooth curves in a complete, ample, base point free linear system $|L|$ on an Enriques surface to the existence of certain secant spaces on the image of the surface mapped by the adjoint system. We also explicitly compute the minimal gonality in terms of invariants of the line bundle $L$. In particular, we obtain a precise criterion for the variation of the gonality of the curves.