Curves on complete intersections and measures of irrationality

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