Deformations with section: Cotangent cohomology, flatness conditions and modular subgerms

We study modular subspaces corresponding to two deformation functors associated to an isolated singularity X_0: the functor Def_{X_0} of deformations of X_0 and the functor Def^s_{X_0} of deformations with section of X_0. After recalling some standard facts on the cotangent cohomology of analytic algebras and the general theory of deformations with section, we give several criteria for modularity in terms of the relative cotangent cohomology modules of a deformation. In particular it is shown that the modular strata for the functors Def_{X_0} and Def^s_{X_0} of quasihomogeneous complete intersection singularities coincide. Flatness conditions for the first cotangent cohomology modules of the deformation functors under consideration are then compared.