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Measures of irrationality for hypersurfaces of large degree

Part of: Curves Cycles and subschemes Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  04 September 2017

Francesco Bastianelli ,
Pietro De Poi ,
Lawrence Ein ,
Robert Lazarsfeld  and
Brooke Ullery
Francesco Bastianelli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bari, Via Edoardo Orabona 4, 70125 Bari, Italy email [email protected]
Pietro De Poi
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università degli Studi di Udine, Via delle Scienze 206, 33100 Udine, Italy email [email protected]
Lawrence Ein
Affiliation:
Department of Mathematics, University Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607, USA email [email protected]
Robert Lazarsfeld
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA email [email protected]
Brooke Ullery
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA email [email protected]
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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 $X\subseteq \mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\geqslant 2n+1$, then any dominant rational mapping $f:X{\dashrightarrow}\mathbf{P}^{n}$ 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.

Keywords

hypersurfacesmeasures of irrationalitypositivitydegree of irrationalitycovering gonality

MSC classification

Primary: 14J70: Hypersurfaces 14E05: Rational and birational maps 14E20: Coverings
Secondary: 14C99: None of the above, but in this section 14H51: Special divisors (gonality, Brill-Noether theory)
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 11 , November 2017 , pp. 2368 - 2393
Copyright
© The Authors 2017 

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