Seshadri positive curves in a smooth projective 3-fold

A notion of positivity, called Seshadri ampleness, is introduced for a smooth curve C in a polarized smooth projective 3-fold (X,A), whose motivation stems from some recent results concerning the gonality of space curves and the behaviour of stable bundles on P3 under restriction to C. This condition is stronger than the normality of the normal bundle and more general than C being defined by a regular section of an ample rank-2 vector bundle. We then explore some of the properties of Seshadri-ample curves.