# Equisingular and Equinormalizable Deformations of Isolated Non-Normal Singularities

## TL;DR

This paper introduces new invariants and criteria for understanding equisingularity and equinormalizability in families of isolated non-normal singularities, providing both theoretical insights and topological characterizations.

## Contribution

It defines $ ext{delta}$ and $ ext{mu}$ invariants for INNS and establishes numerical conditions for equinormalizability, extending classical results and offering a comprehensive survey.

## Key findings

- Necessary and sufficient conditions for equinormalizability using $ ext{delta}$ and $ ext{mu}$
- Determination of the number of connected components of the Milnor fibre
- Topological triviality criteria for families of generically reduced curves

## Abstract

We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using $\delta$ and $\mu$. Moreover, we determine the number of connected components of the Milnor fibre of an arbitrary INNS. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families. Finally we state some open problems and conjectures. In addition we give a survey of classical results about equisingularity and equinormalizability so that the article may be useful as a reference source.