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 universal family of lines $I\to F$ and various subloci. We first describe the moduli and ramification theory of the genus four fibration $p:I\to X$ and explore its relation to a birational model of $F$ in $I$. The main part of the paper is devoted to describing the locus $V\subset F$ of triple lines, i.e., the fixed locus of the Voisin map $\varphi :F⇢F$, in particular proving it is an irreducible projective singular surface of class $21{\mathrm{c}}_2({\mathcal{U}}_F)$ and detailing its intersection with the locus $S$ of second type lines. A consequence of the analysis of the singularities of $V$ is a geometric proof of the fact that if $X$ is very general, then the number of singular (necessarily 1-nodal) rational curves in $F$ of primitive class is 3780.

Minor corrections. Published in IMRN. (This paper was split off from an older version of arXiv:2008.05162)

Cited by in corpus (2)

• Remarks on the geometry of the variety of planes of a cubic fivefold
• On some invariants of cubic fourfolds