Blowups with log canonical singularities

TL;DR

This paper proves a bound on the minimal weight of weighted blow-ups with log canonical singularities in affine space, confirming a conjecture by Birkar and providing explicit bounds in four dimensions.

Contribution

It establishes a universal bound depending only on dimension and singularity type, and explicitly computes bounds for four-dimensional cases with terminal singularities.

Findings

Bound on minimal weight depends only on dimension and epsilon

Explicit bound of 32 for four-dimensional blowups with terminal singularities

Most four-dimensional cases have minimal weight at most 6

Abstract

We show that the minimum weight of a weighted blow-up of $\mathbf A^d$ with $\varepsilon$-log canonical singularities is bounded by a constant depending only on $\varepsilon $ and $d$. This was conjectured by Birkar. Using the recent classification of $4$-dimensional empty simplices by Iglesias-Vali\~no and Santos, we work out an explicit bound for blowups of $\mathbf A^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.