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 $Xsubset mathbb P^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $dge 2n+1$, then any dominant rational mapping $fcolon Xdashrightarrow mathbb P^n$ 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 concerning varieties of multi-secant lines.

Measures of irrationality for hypersurfaces of large degree

DE POI, Pietro;
2017

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 $Xsubset mathbb P^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $dge 2n+1$, then any dominant rational mapping $fcolon Xdashrightarrow mathbb P^n$ 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 concerning varieties of multi-secant lines.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11390/1119040
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