Blowups With Log Canonical Singularities

We show that the minimum weight of a weighted blowup of A(d) with epsilon-log canonical singularities is bounded by a constant depending only on epsilon and d. This was conjectured by Birkar.Using the recent classification of 4-dimensional empty simplices by Iglesias-Valino and Santos, we work out an explicit bound for blowups of A(4) with terminal singularities: the smallest weight is always at most 32, and at most 6 in all but finitely many cases.