A New Proof for the Embedded Resolution of Surface Singularities in Arbitrary Characteristic

TL;DR

This paper presents a novel proof for resolving surface singularities embedded in three-dimensional spaces over fields of any characteristic, using a new invariant that ensures a systematic resolution process.

Contribution

It introduces a new upper semicontinuous resolution invariant inspired by characteristic zero proofs, overcoming issues faced in positive characteristic settings.

Findings

The resolution invariant strictly decreases with each blowup.

The proof employs techniques to maximize invariants over all coordinate changes.

It generalizes classical methods like the Tschirnhausen transformation to positive characteristic.

Abstract

A new proof for the embedded resolution of surface singularities in a three-dimensional smooth ambient space over algebraically closed fields of arbitrary characteristic. The proof makes use of an upper semicontinuous resolution invariant which prescribes the center in each step of the resolution algorithm. The resolution invariant strictly decreases under each blowup. The definition of the resolution invariant is inspired by the proofs of resolution of singularities in arbitrary dimension over fields of characteristic zero. It was developed in an attempt to generalize these proofs to the setting of arbitrary characteristic. The usual resolution invariant which is used in characteristic zero behaves very badly over fields of positive characteristic. It is not upper semicontinuous. Further, the invariant may increase under blowup which destroys the induction argument. It is shown how these problems can be overcome in the case of surface singularities which are embedded in a three-dimensional smooth ambient space. Analogous to the proofs of resolution of singularities in characteristic zero, coefficient ideals are used to enable induction on the dimension of the ambient space. Since hypersurfaces of maximal contact need not exist over fields of positive characteristic, coefficient ideals are considered instead with respect to all regular hypersurfaces. The associated invariants are then maximized over all local flags of smooth subspaces. To construct flags which realize this maximum, techniques are introduced to maximize invariants of coefficient ideals over all coordinate changes. These so-called "cleaning techniques" form a core part of the proof. They can be seen as a characteristic-free generalization of the Tschirnhausen transformation, a classical technique to construct hypersurfaces of maximal contact.