Elliptic and K3 fibrations on double octic Calabi-Yau threefolds

TL;DR

This work analyzes elliptic and K3 fibrations on double octic Calabi–Yau threefolds defined by eight-plane arrangements in $\mathbb{P}^3$, establishing that every double octic admits numerous such fibrations. It constructs elliptic fibrations from high-multiplicity arrangement points and from pairs of skew arrangement lines, and develops three K3 fibration types: Kummer fibrations, double sextic fibrations, and double quadric fibrations, deriving explicit models and invariants for each. A key application shows a birational equivalence between two one-parameter families of double octics by comparing their K3 fibrations via Picard-Fuchs operators, and provides explicit birational maps linking the families. The results enrich the understanding of fibration structures on CY threefolds, offering concrete geometric tools and birational connections among double octic families with potential implications for arithmetic and string-theoretic contexts.

Abstract

We study fibrations by elliptic curves and K3 surfaces of double octic Calabi-Yau threefolds determined by singular lines and points of multiplicity at least 4 of the defining octic arrangement. As a consequence we conclude that every double octic admits many elliptic and K3 fibrations. We apply descriptions of some K3 fibrations to construct a birational map between elements of two families of double octics.

Theorems & Definitions (7)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 5.1