Covering gonalities of complete intersections in positive characteristic

TL;DR

This paper introduces the concepts of covering gonality and separable covering gonality for varieties over arbitrary fields, establishing lower bounds for complete intersections in positive characteristic.

Contribution

It generalizes the notion of covering gonality to positive characteristic and provides new lower bounds for complete intersections over arbitrary fields.

Findings

Separable covering gonality of smooth complete intersections is at least d - N + 1.

Very general complete intersections have covering gonality at least (d - N + 2)/2.

The definitions extend previous complex-analytic concepts to arbitrary fields.

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-Ullery for complex varieties. We show that over an arbitrary field a smooth multidegree $(d_1,\ldots,d_k)$ 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 $\frac{d-N+2}{2}$.