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Extension of the Topological Abel-Jacobi Map for Cubic Threefolds

Yilong Zhang

TL;DR

This work develops a complete framework for the boundary behavior of the topological Abel-Jacobi map on cubic threefolds by analyzing the locus of primitive vanishing cycles $\mathcal{T}_v$ and its Stein/Nash-type compactifications. It identifies the boundary fibers with orbits of Weyl groups generated by $(-2)$-curves on minimal resolutions of cubic surfaces with ADE singularities, linking local monodromy to the $\mathbb{E}_6$ root system and the global monodromy to Weyl subgroups $W(R_e)$. A refined completion $\tilde{\mathcal{T}}_v$ is constructed to accommodate Eckardt phenomena via semistable reduction, enabling an extension of the topological Abel-Jacobi map to $\text{Bl}_0J(X)$ and a finite, controlled fiber structure (notably degree $24$ over Eckardt pencils). The paper also verifies Clemens’ tube-mapping conjecture in the cubic-threefold setting, showing the tube map has maximal (cofinite) image on primitive homology, and connects the geometry of Hilbert schemes of skew lines with theta-divisor geometry to illuminate the compactifications. Collectively, these results provide a precise geometric and topological description of how Hodge-theoretic data degenerates on cubic fourfold hyperplane sections and how the Abel-Jacobi information extends to boundary models, with potential implications for higher-degree hypersurfaces.

Abstract

The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing cycle forms a 72-to-1 covering space $T_v$ of a Zariski open subspace of $(\mathbb P^4)^*$. Based on a lemma of Stein on the compactification of finite analytic covers, we found a compactification of $T_v$ to which the topological Abel-Jacobi map extends. Moreover, the boundary points of the compactification can be interpreted in terms of local monodromy and the singularities on cubic surfaces. We prove the associated map on fundamental groups of topological Abel-Jacobi map is surjective.

Extension of the Topological Abel-Jacobi Map for Cubic Threefolds

TL;DR

This work develops a complete framework for the boundary behavior of the topological Abel-Jacobi map on cubic threefolds by analyzing the locus of primitive vanishing cycles and its Stein/Nash-type compactifications. It identifies the boundary fibers with orbits of Weyl groups generated by -curves on minimal resolutions of cubic surfaces with ADE singularities, linking local monodromy to the root system and the global monodromy to Weyl subgroups . A refined completion is constructed to accommodate Eckardt phenomena via semistable reduction, enabling an extension of the topological Abel-Jacobi map to and a finite, controlled fiber structure (notably degree over Eckardt pencils). The paper also verifies Clemens’ tube-mapping conjecture in the cubic-threefold setting, showing the tube map has maximal (cofinite) image on primitive homology, and connects the geometry of Hilbert schemes of skew lines with theta-divisor geometry to illuminate the compactifications. Collectively, these results provide a precise geometric and topological description of how Hodge-theoretic data degenerates on cubic fourfold hyperplane sections and how the Abel-Jacobi information extends to boundary models, with potential implications for higher-degree hypersurfaces.

Abstract

The difference of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing cycle forms a 72-to-1 covering space of a Zariski open subspace of . Based on a lemma of Stein on the compactification of finite analytic covers, we found a compactification of to which the topological Abel-Jacobi map extends. Moreover, the boundary points of the compactification can be interpreted in terms of local monodromy and the singularities on cubic surfaces. We prove the associated map on fundamental groups of topological Abel-Jacobi map is surjective.
Paper Structure (32 sections, 49 theorems, 76 equations, 14 figures, 1 table)

This paper contains 32 sections, 49 theorems, 76 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Cubic Threefolds
  3. Abel-Jacobi Map and Compactification of $\mathcal{T}_v$
  4. Limiting Primitive Vanishing Cycles and Resolution of ADE Singularities
  5. Extension of Topological Abel Jacobi Map
  6. Tube Mapping
  7. Preliminaries
  8. Root System on Cubic Surfaces
  9. Locus of Primitive Vanishing Cycles
  10. Eckardt Points
  11. Abel-Jacobi Map
  12. Hilbert Scheme of a Pair of Skew Lines
  13. Compactification of the Primitive Vanishing Cycle Component
  14. Stein's Compactification
  15. Nash Blow-up
  16. ...and 17 more sections

Key Result

Proposition 1.4

(cf. Theorem barT'BlowupThm) There is birational morphism $\textup{Bl}_0(\Theta)\to \bar{\mathcal{T}}_v$, which contracts finitely many elliptic curves corresponding to the Eckardt points on the cubic threefold $X$.

Figures (14)

  • Figure 1: $\mathbb E_6$ Dynkin Diagram
  • Figure 2: Schemes of the Four Types
  • Figure :
  • Figure :
  • Figure :
  • ...and 9 more figures

Theorems & Definitions (93)

  • Proposition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Conjecture 1.10
  • Proposition 1.11
  • Definition 2.1
  • Example 2.2
  • ...and 83 more