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.

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 János Kollár on the occasion of his sixtieth birthday

References in corpus (1)

• Unirational threefolds with no universal codimension 2 cycle

Cited by in corpus (7)

• On irrationality of surfaces in ${\mathbb{P}}^3$
• Degree of irrationality of very general abelian surfaces
• Gonality of curves on general hypersurfaces
• Fano hypersurfaces with arbitrarily large degrees of irrationality
• On the degree of irrationality of low genus $K3$ surfaces
• Positivity Results for spaces of rational curves
• Covering gonalities of complete intersections in positive characteristic