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 PN 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.