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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Measures of irrationality of the Fano surface of a cubic threefold


Authors: Frank Gounelas and Alexis Kouvidakis
Journal: Trans. Amer. Math. Soc. 371 (2019), 7111-7133
MSC (2010): Primary 14M20, 14H10, 14J45; Secondary 14J30, 14N20, 14M15
DOI: https://doi.org/10.1090/tran/7565
Published electronically: October 17, 2018
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Abstract: For $ X$ a smooth cubic threefold, we study the Plücker embedding of the Fano surface of lines $ S$ of $ X$. We prove that if $ X$ is general, then the minimal gonality of a covering family of curves of $ S$ is four, and that this happens for a unique family of curves. The analysis also shows that there is a unique pentagonal connecting family of curves, which leads to the fact that the connecting gonality of $ S$ is five, whereas the degree of irrationality, i.e., the minimal degree of a rational map from $ S$ to $ \mathbb{P}^2$, is six.


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Additional Information

Frank Gounelas
Affiliation: Mathematisches Institut, Humboldt Universität Berlin, 10099 Berlin, Germany
Email: gounelas@mathematik.hu-berlin.de

Alexis Kouvidakis
Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, 70013 Heraklion, Greece
Email: kouvid@uoc.gr

DOI: https://doi.org/10.1090/tran/7565
Keywords: Cubic threefold, Fano scheme of lines, covering gonality, degree of irrationality
Received by editor(s): July 28, 2017
Received by editor(s) in revised form: January 22, 2018, and February 21, 2018
Published electronically: October 17, 2018
Article copyright: © Copyright 2018 American Mathematical Society