Measures of irrationality of the Fano surface of a cubic threefold

Gounelas, Frank; Kouvidakis, Alexis

doi:10.1090/tran/7565

Measures of irrationality of the Fano surface of a cubic threefold
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by Frank Gounelas and Alexis Kouvidakis PDF

Trans. Amer. Math. Soc. 371 (2019), 7111-7133 Request permission

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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