On a generic symmetry defect hypersurface
  Let f : X -> Y be a dominant polynomial mapping of affine varieties. For generic y in Y we have Sing(f^{-1}(y)) = f^{-1}(y) \cap Sing(X): As an application we show that symmetry defect hypersurfaces for two generic members of the irreducible algebraic family of n-dimensional smooth irreducible subvarieties in general position in C^{2n} are homeomorphic and they have homeomorphic sets of singular points. In particular symmetry defect curves for two generic curves in C^2 of the same degree have the same number of singular points. 
S. Janeczko, Z. Jelonek, M.A.S. Ruas
math.AG