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

Abstract

We study Sarkisov links initiated by the toric weighted blowup of a point in ${\mathbb{P}}^4$ 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.