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

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We study modular subspaces corresponding to two deformation functors associated to an isolated singularity X₀: the functor Defₓ₀ of deformations of X₀ and the functor Defˢₓ₀ of deformations with section of X₀. 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 Defₓ₀ and Defˢₓ₀ of quasihomogeneous complete intersection singularities coincide. Flatness conditions for the first cotangent cohomology modules of the deformation functors under consideration are then compared.