The Anatomy of a Singularity

1. Some basic facts Denote by O = ON+1 the ring of germs of holomorphic functions f = f(z0, · · · , zN ) defined in a neighborhood of ~0 ∈ CN+1. We denote by m ⊂ O the maximal ideal of O, f ∈ m ⇐⇒ f(~0) = 0. Let f ∈ m. Assume ~0 is an isolated critical point of f , i.e. ~0 is an isolated point of the variety ∂zif = 0, ∀i = 0, · · · , N. We define the Jacobian ideal of f to be the ideal Jf ⊂ O generated by ∂zif , i = 0, · · · , N . From the analytical Nullstellensatz we deduce √ Jf = m ⇐⇒ ∃k > 0 : m ⊂ Jf ⇐⇒ Af := dimCO/Jf < ∞. The finite dimensional commutative C -algebra Af is called the local algebra of the critical point ~0 of f . Its dimension is called the Milnor number of f at ~0 and it is denoted by μ = μ(f,~0). It has a natural structure of C{t}-algebra t · (g mod Jf ) = (fg) modJF , ∀g ∈ O. For every positive integer N we denote by jN (f) the N -th jet of f . It can be identified with a polynomial of degree N in n + 1 complex variables. Two germs f, g ∈ m are called right-equivalent and we write this f ∼r g if g is obtained from g by a change in variables.