Blowups with log canonical singularities

Abstract

We show that the minimum weight of a weighted blowup of ${\mathbb{𝔸}}^d$ with $𝜀$–log canonical singularities is bounded by a constant depending only on $𝜀$ and $d$. This was conjectured by Birkar.

Using the recent classification of $4$–dimensional empty simplices by Iglesias-Valiño and Santos, we work out an explicit bound for blowups of ${\mathbb{𝔸}}^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.