The movable cone of Calabi--Yau threefolds in ruled Fano manifolds

TL;DR

This paper explicitly describes the chamber structure of the movable cone for certain Calabi--Yau threefolds in ruled Fano manifolds and identifies all their birational minimal models, showing finiteness.

Contribution

It provides a detailed description of the movable cone's chamber structure and classifies all birational minimal models for these Calabi--Yau threefolds, establishing their finiteness.

Findings

Explicit chamber structure of the movable cone

Finite number of birational minimal models

Complete classification in the specified setting

Abstract

We describe explicitly the chamber structure of the movable cone for a general complete intersection Calabi--Yau threefold in a non-split $(n + 4)$-dimensional $\mathbb{P}^{n}$-ruled Fano manifold of index $n + 1$ and Picard number two. Moreover, all birational minimal models of such Calabi--Yau threefolds are found whose number is finite.