Sarkisov links from toric weighted blowups of $\mathbb{P}^3$ and $\mathbb{P}^4$ at a point

TL;DR

This paper classifies and explicitly describes Sarkisov links initiated by toric weighted blowups of points in projective spaces, providing criteria for when these blowups are weak Fano, using variation of GIT.

Contribution

It offers a complete classification of Sarkisov links from toric weighted blowups in projective spaces and characterizes weak Fano conditions via weights.

Findings

Classified which toric weighted blowups initiate Sarkisov links.

Explicit descriptions of the Sarkisov links are provided.

A simple weight-based criterion for weak Fano property is established.

Abstract

We study Sarkisov links initiated by the toric weighted blowup of a point in $\mathbb{P}^3$ or $\mathbb{P}^4$ using variation of GIT. We completely classify which of these initiate Sarkisov links and describe the links explicitly. Moreover, if $X$ is the toric weighted blowup of $\mathbb{P}^d$ at a point, we give a simple criterion in terms of the weights of the blowup that characterises when $X$ is weak Fano.