Measures of irrationality for hypersurfaces of large degree

TL;DR

This paper investigates measures of irrationality for high-degree hypersurfaces, establishing lower bounds on rational coverings and gonality, and connecting positivity of canonical bundles to these invariants.

Contribution

It provides new lower bounds for rational maps from hypersurfaces of large degree and offers simplified proofs of existing results using positivity properties.

Findings

Dominant rational maps from hypersurfaces have degree at least d-1.

Positivity of canonical bundles yields lower bounds on irrationality measures.

New proofs of results by Ran and Beheshti-Eisenbud are presented.

Abstract

We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if X is a very general smooth hypersurface of dimension n and degree d \ge 2n+1, then any dominant rational mapping from X to projective n-space must have degree at least d-1. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti-Eisenbud.