# Deformations of corank $1$ frontals

## TL;DR

This paper develops a new theoretical framework for understanding deformations of corank 1 frontals, including stability, versality, and classification, without relying on contact geometry, and proves a version of Mond's conjecture.

## Contribution

It introduces a frontal Thom-Mather theory, characterizes stability and versality, and classifies stable corank 1 frontals in complex dimensions.

## Key findings

- Proves a frontal version of Mond's conjecture in dimension 1.
- Provides a complete classification of stable corank 1 frontals from a0a0 to a0a0.
- Develops methods for constructing stable and versal frontal unfoldings.

## Abstract

We develop a Thom-Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather-Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb C^3$ to $\mathbb C^4$.