Curves on complete intersections and measures of irrationality (with an appendix by Mohan Swaminathan)

We study the minimal degrees and gonalities of curves on complete intersections. We prove that the degree of any curve on a general complete intersection X \subseteq \mathbb{P}^{N} of large multidegree is bounded from below by the degree of X . As an application, we answer a problem of Bastianelli–De Poi–Ein–Lazarsfeld–Ullery on measures of irrationality for complete intersections.