Effective resolution of singularities

Abstract

Consider a projective variety $X\subset {\mathbb{P}}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E\subset {\mathbb{P}}^n$, where the degrees of both $X$ and $E$ are at most $d$. We show there is a pair $(n^{\prime },d^{\prime })$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities $(X^{\prime },E^{\prime })$, where $(X^{\prime },E^{\prime })$ can be embedded in ${\mathbb{P}}^{n^{\prime }}$ and both $X^{\prime }$ and $E^{\prime }$ have degrees at most $d^{\prime }$ in ${\mathbb{P}}^{n^{\prime }}$.

22 pages, to appear in Pure Appl. Math. Quarterly, volume in honour of Caucher Birkar