The Clemens Conjectures for Cubic Threefolds relative to a Hyperplane

TL;DR

This work extends the Clemens conjectures from absolute Calabi–Yau threefolds to the relative, 1/2-log Calabi–Yau setting given by a cubic threefold $X$ and a hyperplane section $Y$. By exploiting a $\frac{1}{2}$-log deformation–obstruction duality, it translates the injectivity of the relative infinitesimal Abel–Jacobi map into concrete predictions about anchored rational curves, predicting finiteness and a prescribed normal bundle $N_{C/X}(-Y)\cong O(-1)\oplus O(-1)$ for moving curves with fixed intersection on $Y$. The paper organizes this into absolute and relative analyses, connects to limiting Hodge structures, and frames the relative problem within degenerations and mirror-symmetry perspectives. The results provide a roadmap for counting rational curves in the relative setting and offer a coherent link between log geometry, Noether–Lefschetz phenomena, and degeneration techniques.

Abstract

We propose an analogue of the Clemens conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds, specifically for the pair $(X, Y)$ where $X$ is a cubic threefold and $Y$ is a hyperplane section. By exploiting a perfect deformation/obstruction duality specific to the $\frac{1}{2}$-log setting, we formulate conjectures regarding the injectivity of the relative infinitesimal Abel-Jacobi map and the finiteness of rational curves with fixed intersection on $Y$.

Theorems & Definitions (8)

• Conjecture 2.1: II, Clemens
• Theorem 2.1
• Corollary 1
• Theorem 3.1
• proof
• Theorem 3.2
• Corollary 2
• Conjecture 3.1