Effective resolution of singularities

TL;DR

The article proves that, for a projective variety $X$ and a reduced snc divisor $E$ with degrees bounded by $d$, there exist computable embedding and degree bounds $(n',d')$ ensuring a log resolution $(X',E')$ with $\deg X' , \deg E' \le d'$ in $\mathbb{P}^{n'}$. Resolution is achieved by functorial blowings-up guided by marked ideals, reducing complex embedded desingularization to the resolution of these marked objects and their derivative/coefficient data on maximal contact hypersurfaces. The authors provide explicit, cascading degree and dimension bounds, culminating in Grzegorczyk-class complexity estimates for the full desingularization process. This work establishes effective, implementable bounds on the resolution procedure, highlighting that ambient-dimension growth dominates degree growth, and furnishes a framework compatible with algorithmic implementations in algebraic geometry.

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 $\mathbb{P}^{n'}$.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Theorem 2.4
• proof
• Example 3.2