Effective resolution of singularities

TL;DR

The paper provides an explicit method to compute bounds on the degrees and dimensions needed to resolve singularities of projective varieties with normal crossings divisors over algebraically closed fields of characteristic zero.

Contribution

It introduces an explicit, computable pair (n',d') that bounds the degrees and dimensions of the resolved variety and divisor after resolution.

Findings

Explicit bounds (n',d') for resolution are derived.

Resolution process preserves degree bounds in projective space.

Method applies to varieties with simple normal crossings divisors.

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 $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities $(X',E')$, where $(X',E')$ can be embedded in $\mathbb{P}^{n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\mathbb{P}^{n'}$.