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FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  08 May 2020

NATHAN CHEN  and
DAVID STAPLETON
NATHAN CHEN
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY11794, USA; nathan.chen@stonybrook.edu
DAVID STAPLETON
Affiliation:
Department of Mathematics, UC San Diego, La Jolla, CA92093, USA; dstapleton@ucsd.edu

Abstract

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We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$, then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

MSC classification

Primary: 14E05: Rational and birational maps
Secondary: 14J45: Fano varieties
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e24
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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