Deformations of plane curve singularities and the δ-constant stratum

Consider the germ of a plane curve (C0, 0) := V (f) c C2 with an isolated singularity at 0 where f 2 OC2,0. The δ-invariant of (C0, 0) can be interpreted as the maximum number of singularities that can pile up on the zero level set of a deformation of f. Let F 2 OC2xC";0 be a miniversal deformation of f then the δ-constant stratum D(δ) in the discriminant of F is the set of parameters where the δ-invariant of the deformed curve is equal to the δ-invariant of the original curve. Givental and Varchenko showed that when (C0, 0) is irreducible, then D(δ) is an example of a Lagrangian singularity with respect to a symplectic form arising from the intersection pairing on the deformed curves. More recently van Straten and Sevenheck have developed a theory of deformations of Lagrangian singularities and conjecture that D(δ) is a rigid Lagrangian singularity when (C0, 0) is an irreducible plane curve singularity. In this thesis we will show how to compute the symplectic form explicitly in the case of an irreducible simple singularity. Using this symplectic form we construct a maximal Cohen-Macaulay module on the discriminant that can be used to find equations for D(δ) for the A2k,E6 and E8 singularities. We will add weight to the conjecture of van Straten and Sevenheck by showing that D(δ) is Cohen-Macaulay for E6 and E8