Geometry of lines on a cubic fourfold

Abstract

For a general cubic fourfold $X\subset\mathbb{P}^5$ with Fano scheme of lines $F$, we prove a number of properties of the fibration of genus 4 curves from the universal family of lines $p:I\to X$. We compute the classes of various ramification loci attached to this fibration and use this to compute the class of the locus of triple lines, i.e., the fixed locus $V$ of the Voisin map $\phi:F\dashrightarrow F$, which we prove is a smooth irreducible surface if $X$ is general. In the final two sections, we compute the Hodge numbers of the locus $S\subset F$ of lines of second type and give an upper bound for the degree of irrationality of the Fano scheme of lines of any smooth cubic hypersurface