Circle of Sarkisov links on a Fano $3$-fold

TL;DR

This paper constructs two new birational models of a general Fano 3-fold within the Sarkisov program, illustrating the concept of relations as a circle of Sarkisov links, and enhancing understanding of birational geometry.

Contribution

It introduces two novel birational models of a Fano 3-fold and visualizes their relations as a circle of Sarkisov links, advancing the study of birational relations.

Findings

Construction of two new Mori fibre space models

Visualization of Sarkisov link relations as a circle

Enhanced understanding of birational relations in Fano 3-folds

Abstract

For a general Fano $3$-fold of index $1$ in the weighted projective space $\mathbb{P}(1,1,1,1,2,2,3)$ we construct $2$ new birational models that are Mori fibre spaces, in the framework of the so-called Sarkisov program. We highlight a relation between the corresponding birational maps, as a circle of Sarkisov links, visualising the notion of relations (due to Kaloghiros) in Sarkisov program.