Blowups with log canonical singularities

Abstract

Using only elementary toric methods and facts about quotient singularities and lattice simplices, we show that for any fixed dimension d there is a constant ℓ d such that if a weighted blowup of A d has only canonical singularities then the smallest of the weights cannot exceed ℓ d . This is a special case of a conjecture of Birkar.Using the recent classification of 4-dimensional empty simplices by Iglesias-Valiño and Santos, we show that for blowups of A 4 with terminal singularities the smallest weight is at most 32, and at most 6 in all but finitely many cases.