Covering gonality of hypersurfaces in a product of projective spaces

TL;DR

This paper studies the covering gonality of very general hypersurfaces in products of projective spaces, showing it can be computed via fibered families and establishing bounds for large bi-degree cases.

Contribution

It extends the understanding of covering gonality from projective spaces to products, providing methods to compute it and bounds for large bi-degree hypersurfaces.

Findings

Covering gonality can be computed through fibered hypersurface families.

Curves computing gonality lie within fibers, not transversally.

Established lower bounds for joint covering gonality in large bi-degree cases.

Abstract

In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case of a hypersurface in a projective space, we establish a similar result for very general smooth hypersurfaces of sufficiently large bi-degree. More precisely, we show that the covering gonality of such hypersurfaces can be computed by viewing them as a family of hypersurfaces over a projective space. Then, the curves computing the covering gonality lie entirely within the fibres of one of the families. This rules out transversal curves from computing the covering gonality. In addition to this, we investigate the behaviour of the joint covering gonality as in [LM23] and establish a lower bound for bi-degree large enough.