On deformations of maps and curve singularities

TL;DR

This paper provides explicit formulas for deformation functors related to curve singularities, linking classical invariants to the smoothness and dimension of local moduli spaces, and estimates for the A_e-codimension of parametrized singularities.

Contribution

It introduces new explicit formulas for cotangent cohomology groups of deformation functors associated with curve singularities, enhancing understanding of their moduli spaces.

Findings

Formulas for cotangent cohomology groups in terms of classical invariants

Criteria for smoothness and dimension of local moduli spaces

Explicit estimates for A_e-codimension of parametrized curve singularities

Abstract

We study several deformation functors associated to the normalization of a reduced curve singularity $(X,0) \subset (\c^n,0)$. The main new results are explicit formulas, in terms of classical invariants of (X,0), for the cotangent cohomology groups $T^i, i = 0,1,2,$ of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the $\hoa{A}_e$-codimension of a parametrized curve singularity, where $\hoa{A}_e$ denotes the Mather-Wall group of left-right equivalence.