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

TL;DR

The paper presents a method for resolving singularities of plane curves using weighted blow-ups and torus actions, avoiding restrictions on the base field.

Contribution

It introduces an improved resolution technique employing weighted blow-ups and torus actions, extending previous methods to more general settings.

Findings

Constructed embedded resolutions via weighted blow-ups.

Utilized torus actions for inductive resolution steps.

Achieved resolution without restrictions on the base field.

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.