Sarkisov links from toric weighted blowups of \(\mathbb{P}^4\) at a point (Q2694629)

Summary: 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.