Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics

Elena Sammarco

TL;DR

The paper constructs and analyzes a new irreducible Severi divisor in the moduli space of cubic fourfolds, consisting of fourfolds that admit a polar quadric net with a discriminant equal to a 10-nodal plane sextic. It develops a framework based on Dixon's lemma, spin-curve theory, and Severi varieties to produce a Severi hypersurface $\mathcal{D}_{sn}$ in the cubic fourfold moduli, shows its irreducibility, and proves it is nonspecial (not a Noether-Lefschetz divisor). The argument blends a universal Grassmannian construction, modular reduction, and deformation theory, including a detailed local analysis at the Fermat fourfold to exclude NL-type behavior. Consequently, the work identifies five nonspecial irreducible divisors in the moduli space, enriching the birational geometry of cubic fourfolds and contributing to the understanding of Hodge loci beyond classical NL divisors.

Abstract

In the moduli space $\mathcal{C}$ of complex cubic hypersurfaces $X\subset\mathbb{P}^5$, we study the condition that $X$ admits a net of polar quadrics whose discriminant locus is a $10$-nodal irreducible plane sextic curve. Our main result is that such a condition defines an irreducible divisor in $\mathcal{C}$ which is not of Noether-Lefschetz type.

A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics

TL;DR

The paper constructs and analyzes a new irreducible Severi divisor in the moduli space of cubic fourfolds, consisting of fourfolds that admit a polar quadric net with a discriminant equal to a 10-nodal plane sextic. It develops a framework based on Dixon's lemma, spin-curve theory, and Severi varieties to produce a Severi hypersurface in the cubic fourfold moduli, shows its irreducibility, and proves it is nonspecial (not a Noether-Lefschetz divisor). The argument blends a universal Grassmannian construction, modular reduction, and deformation theory, including a detailed local analysis at the Fermat fourfold to exclude NL-type behavior. Consequently, the work identifies five nonspecial irreducible divisors in the moduli space, enriching the birational geometry of cubic fourfolds and contributing to the understanding of Hodge loci beyond classical NL divisors.

Abstract

In the moduli space of complex cubic hypersurfaces , we study the condition that admits a net of polar quadrics whose discriminant locus is a -nodal irreducible plane sextic curve. Our main result is that such a condition defines an irreducible divisor in which is not of Noether-Lefschetz type.
Paper Structure (12 sections, 61 equations)

This paper contains 12 sections, 61 equations.

Table of Contents

  1. Introduction
  2. The Dixon's lemma
  3. Theta characteristics on smooth irreducible curves
  4. Compactification of $\mathcal{S}_g$
  5. The Dixon's lemma
  6. Construction of the Severi divisor
  7. The polar linear system of a cubic fourfold
  8. Cubic fourfolds with the same polar net
  9. The Severi varieties of plane sections of $\Delta_5$
  10. The Severi hypersurface $\mathsf{D}_{sn}$ in $\mathbb P^{55}$
  11. Modular constructions
  12. The Severi divisor is nonspecial

Theorems & Definitions (12)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 2 more