Singularities of Varieties

Infinitesimal Stability
Let D m,n be the set of germs of differentiable functions f : (R m , 0) →
(R n , 0). Let G m be the group of germs of automorphisms of (R m ,0).
Obviously G m × G n acts on D m,n but this is not enough.
Take f ∈ D m,n represented by a function defined on U ⊂ R m . For any
x 0 ∈ U, x → (f(x + x 0 ) − f(x 0 )) is also in D m,n . For fixed U, this defines
an action, and the derivative of this action at 0 does not depend on the
choice of U.
We say that f is infinitesimally stable iff the differential of the action above
(isomorphism and translation together) is surjective.
VO AAG 29