Measures of irrationality of the Fano surface of a cubic threefold

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.

arXiv comment21 pages, 1 figure. Minor corrections, final version. To appear in Transactions of the AMS