Measures of irrationality for hypersurfaces of large degree
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.
Keywords
Cite
@article{arxiv.1511.01359,
title = {Measures of irrationality for hypersurfaces of large degree},
author = {Francesco Bastianelli and Pietro De Poi and Lawrence Ein and Robert Lazarsfeld and Brooke Ullery},
journal= {arXiv preprint arXiv:1511.01359},
year = {2019}
}
Comments
Major revision of first version, combining it with previously separate appendix of Bastianelli and De Poi. Extended section of open problems added, as well as new proofs of results of Ran and Beheshti-Eisenbud. Dedicated to J\'anos Koll\'ar on the occasion of his sixtieth birthday