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We study modular subspaces corresponding to two deformation functorsassociated to an isolated singularity X_0: the functor Def_{X_0} ofdeformations of X_0 and the functor Def^s_{X_0} of deformations with section ofX_0. After recalling some standard facts on the cotangent cohomology ofanalytic algebras and the general theory of deformations with section, we giveseveral criteria for modularity in terms of the relative cotangent cohomologymodules of a deformation. In particular it is shown that the modular strata forthe functors Def_{X_0} and Def^s_{X_0} of quasihomogeneous completeintersection singularities coincide. Flatness conditions for the firstcotangent cohomology modules of the deformation functors under considerationare then compared.

Deformations with Section: Cotangent Cohomology, Flatness Conditions, and Modular Subgerms