Teissier singularities

TL;DR

This work addresses resolution of singularities in positive characteristic by introducing Teissier singularities as a positive-characteristic counterpart to quasi-ordinary singularities, defined via the invariant $kappa(pi)$ and its last component. It develops a deformation-theoretic framework where Teissier singularities arise as special fibers in equisingular families whose generic fiber is quasi-ordinary, enabling a simultaneous embedded resolution by a toric morphism. The construction relies on the projected and weighted Hironaka polyhedra to define $kappa(pi)$, overweight deformations, and ghost monomials. The results provide a Jung-like strategy for resolution in characteristic $p$ by relating to characteristic $0$ via toric methods and offer new insight into the link between discriminant and Teissier criteria.

Abstract

The goal of this note is to introduce Teissier singularities and to explain why they are candidate to play, in positive characteristics, a role for resolution of singularities which is similar to the role played by quasi-ordinary singularities in characteristic zero.

Theorems & Definitions (11)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Example 2.1
• Theorem 2.2
• Definition 2.3
• Example 2.4
• Proposition 3.1
• proof
• Theorem 3.2