Logarithmic resolution of singularities in characteristic 0 using weighted blow-ups

TL;DR

This work delivers a constructive, functorial logarithmic resolution of singularities in characteristic zero by marrying ATW-weighted resolution with Quek's toroidal framework. Central to the method is the extended logarithmic invariant $\operatorname{loginv}^*$ (and its refined version $\operatorname{loginv}^*_\,$) guiding a finite sequence of smooth weighted blow-ups that produce a log-smooth ambient with a simple normal crossings divisor while preserving functoriality under log smooth morphisms. The paper develops a comprehensive principalization theory in both smooth and log-smooth settings, introducing derivatives, orders, coefficient ideals, and invariant-based centers, and provides explicit local and global descriptions of the blow-up process. The resulting algorithm yields a canonical, termination-guaranteed path to principalize total transforms with SNC geometry, with significant implications for birational geometry in characteristic zero.

Abstract

In characteristic zero, we construct logarithmic resolution of singularities, with simple normal crossings exceptional divisor, using weighted blow-ups.

Figures (1)

• Figure 1: Two viewpoints of $B$: on the left, the degeneration to the normal cone of $Y$. On the right, a view from the northeast shows it as a birational cobordism Wlodarczyk-birational-cobordism, a link between $Y$ and its blowup. The vertical projection can be viewed as the momentum map $(|s|^2 - \sum w_i |x_i'|^2)/2$, but we digress.

Theorems & Definitions (35)

• Theorem 1.1.1: Hironaka, 1964
• Theorem 1.4.1: Functorial logarithmic resolution of singularities
• Remark 1.4.2
• Remark 1.4.3
• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Proposition 2.1.4
• Definition 2.2.1: See Kollar
• Proposition 2.2.2