Covering gonalities of complete intersections in positive characteristic

We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli-de Poi-Ein-Lazarsfeld-Ullery for complex varieties. We show that over an arbitrary 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 variety has covering gonality at least $\frac{d-N+2}{2}$.