Covering gonalities of complete intersections in positive characteristic

Abstract

We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery for complex varieties. We show that, over an algebraically closed field, a smooth multidegree $(d_1,\dots ,d_k)$ complete intersection in ${\mathbb{P}}^N$ has separable covering gonality at least $d-N+1$, where $d=d_1+\cdots +d_k$. We also show that the very general such hypersurface has covering gonality at least $\frac{1}{2}(d-N+2)$.