Fano manifolds of index n-1 and the cone conjecture

TL;DR

This paper verifies the Morrison-Kawamata cone conjecture for the blow-up of certain Fano manifolds of index n-1 along a specific base locus, contributing to the understanding of automorphism group actions on cones.

Contribution

It proves the Morrison-Kawamata cone conjecture for a class of Fano manifolds of index n-1 after blowing up along a general base locus.

Findings

Confirmed the cone conjecture for blow-ups of Fano manifolds of index n-1

Identified conditions under which the automorphism group acts with a rational polyhedral fundamental domain

Extended the conjecture's verification to new classes of Calabi-Yau pairs

Abstract

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, \Delta)$ have finite, rational polyhedral fundamental domains. Let $Z$ be an $n$-dimensional Fano manifold of index $n-1$ such that $-K_Z = (n-1) H$ for an ample divisor $H$. Let $\Gamma$ be the base locus of a general $(n-1)$-dimensional linear system $V \subset |H|$. In this paper, we verify the Morrison-Kawamata cone conjecture for the blow-up of $Z$ along $\Gamma$.