On deformations of maps and curve singularities

Abstract

We study several deformation functors associated to the normalization of a reduced curve singularity (X, 0) ⊂ (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 Ae-codimension of a parametrized curve singularity, where Ae denotes the Mather-Wall group of left-right equivalence.

Lifting free divisors

“…In particular, if both X and S are smooth, then the dual Zariski-Jacobi sequence truncates to a resolution of T 1 X/S , with T 0 X/S as a second syzygy module. In the language of the Thom-Mather theory of the singularities of differentiable maps, T 1 X/S is isomorphic as a vector space to the extended normal space of ϕ under right equivalence, while the cokernel of δ KS : T 0 S → T 1 X/S is isomorphic as a vector space to the extended normal space of ϕ under left-right equivalence (see [13]).…”

Equisingular calculations for plane curve singularities

“…This proof has been known by the second author since about fifteen years and was communicated at several conferences. A preliminary preprint [GR1], joint with Sevin Recillas, has even been cited by some authors. Later on, these results have been extended to positive characteristic in a joint preprint of the authors [CGL] where, in addition, an algorithm to compute the equisingularity stratum was developed and used to prove one of the main results.…”