Brill-Noether special cubic fourfolds of discriminant 14

TL;DR

This paper investigates the Brill-Noether theory of curves on K3 surfaces linked to cubic fourfolds of discriminant 14, revealing conditions for containing disjoint planes and characterizing special loci in the moduli space.

Contribution

It establishes a connection between Brill-Noether special K3 surfaces of degree 14 and the geometric properties of cubic fourfolds, particularly regarding disjoint planes.

Findings

Any smooth curve in the polarization class has maximal Clifford index.

A cubic fourfold contains disjoint planes iff it admits a Brill-Noether special associated K3 surface of degree 14.

The complement of the pfaffian locus in the discriminant 14 divisor is contained in the locus of fourfolds with two disjoint planes.

Abstract

We study the Brill-Noether theory of curves on K3 surfaces that are Hodge theoretically associated to cubic fourfolds of discriminant 14. We prove that any smooth curve in the polarization class has maximal Clifford index and deduce that a cubic fourfold contains disjoint planes if and only if it admits a Brill-Noether special associated K3 surface of degree 14. As an application, the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of discriminant 14 in the moduli space of cubic fourfolds, is contained in the irreducible locus of cubic fourfolds containing two disjoint planes.