  We study certain mild degenerations of algebraic varieties which appear in
the analysis of a large class of supersymmetric theories, including superstring
theory. We analyze Witten's sigma-model and find that the non-transversality of
the superpotential induces a singularization and stratification of the ground
state variety. This stratified variety (the union of the singular ground state
variety and its exo-curve strata) admit homology groups which, excepting the
middle dimension, satisfy the "Kahler package" of requirements, extend the
"flopped" pair of small resolutions to an "(exo)flopped" triple, and is
compatible with mirror symmetry and string theory. Finally, we revisit the
conifold transition as it applies to our formalism.
