Torus actions, weighted blow-ups, and desingularization of plane curves

2025/07/01 by Abramovich, Dan, Quek, Ming Hao, Schober, Bernd
#14B05 #14D23 #14H20 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.01232

Abstract

Given a singular hypersurface in a regular 2-dimensional scheme essentially of finite type over a field, we construct an embedded resolution of singularities by weighted blow-ups. This differs from our earlier work which required multi-weighted blow-ups. We deduce an inductive argument, despite the fact that higher dimensional tangent spaces arise, by taking torus actions and equivariant centers into account. In addition, we do not have to restrict to perfect base fields.

Citations

• Resolving plane curves using stack-theoretic blow-ups
• Constancy of the Hilbert-Samuel function
• Logarithmic resolution via multi-weighted blow-ups
• Resolving singularities of curves with one toric morphism
• Logarithmic resolution via weighted toroidal blow-ups
• Invariance of Hironaka's characteristic polyhedron
• A polyhedral characterization of quasi-ordinary singularities
• Resolution of Singularities of Arithmetical Threefolds II
• Characteristic polyhedra of singularities without completion -- Part II
• Fibonacci numbers and self-dual lattice structures for plane branches
• Characteristic polyhedra of singularities without completion
• Resolution of Singularities
• Geometry of plane curves via Tschirnhausen resolution tower

Related

• Resolving plane curves using stack-theoretic blow-ups
• Streamlining resolution of singularities with weighted blow-ups
• K-theory of Weighted Blowups
• Desingularization of toric and binomial varieties
• Desingularization by blowings-up avoiding simple normal crossings