The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties

TL;DR

This work analyzes the intermediate Jacobian fibration $\mathcal{J}\to\mathcal{U}$ of a general cubic fourfold $X$ containing a plane by embedding it as a finite-cover Lagrangian subfibration of a Beauville–Mukai system on the K3 surface $S$ associated to $X$. The authors develop a general method to construct Lagrangian fibrations in Prym varieties as subfibrations of Beauville–Mukai systems over loci of nodal curves, and then realize $\mathcal{J}$ (up to a degree-$2$ cover) inside the moduli space $M=M_S(0,[5H_S],\chi)$ via a Prym subfibration $P\to\mathcal{V}$. The key contributions include the description of a smooth symplectic subvariety $N$ with a Lagrangian fibration over a Severi-type locus of nodal curves, the identification of the Prym subfibration $P$ with a fixed locus under a symplectic involution, and the explicit geometric setup for cubic fourfolds containing a plane (including a genus-$11$ curve $C_H$ and a $15$-nodal curve on $S$) that connects to OG10-type hyperkähler geometry. These results provide a bridge between intermediate Jacobian fibrations, Prym varieties, and Beauville–Mukai systems, offering new avenues for compactifications and potential extensions via twisted sheaves.

Abstract

We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold $X$ containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to $X$ up to a cover. To do so, we propose a general construction of Lagrangian fibrations in Prym varieties as subfibrations of Beauville-Mukai systems over some loci of nodal curves in linear systems on K3 surfaces.

Theorems & Definitions (56)

• Theorem : = Theorem \ref{['ThmMainPrymSubFibration']}
• Remark 1.1
• Theorem : =\ref{['ThmSubfibrationJacobianNormalizations']}
• Theorem : =\ref{['ThmPrymSubfibrationOverNodalCurves']}
• Theorem 2.1
• Remark 2.2
• Theorem 2.3: AltmanIarrobinoKleimanIrredCompJac
• Theorem 2.4: KleppePhDThesisPicSchemeCurveComp, Theorem 3.2.1
• Remark 2.5
• Theorem 2.6: KleppePhDThesisPicSchemeCurveComp, Lemma 7.2.1