Extremal Contraction of Projective Bundles

TL;DR

This paper studies extremal contractions of projective bundles over Fano varieties, providing new geometric constructions and analyzing cones of divisors, with applications to birational geometry and flips.

Contribution

It introduces a geometric construction of the rooftop flip and extends recent results to higher dimensions for certain projective bundles.

Findings

Constructed examples of projective bundles with smooth blow-up structures.

Computed nef and pseudoeffective cones for all globally generated bundles over projective space with first Chern class 2.

Provided analogues of recent higher-dimensional results of Vats.

Abstract

In this article, we explore the extremal contractions of several projective bundles over smooth Fano varieties of Picard rank $1$. We provide a whole class of examples of projective bundles with smooth blow-up structures, derived from the notion of drums which was introduced by Occhetta-Romano-Conde-Wi\'sniewski to study interaction with $\mathbb{C}^*$-actions and birational geometry. By manipulating projective bundles, we give a simple geometric construction of the rooftop flip, which was introduced recently by Barban-Franceschini. Additionally, we obtain analogues of some recent results of Vats in higher dimensions. The list of projective bundles we consider includes all globally generated bundles over projective space with first Chern class $2$. For each of them, we compute the nef and pseudoeffective cones.