Ideals associated to deformations of singular plane curves

We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus $ES$ which parametrizes equisingular or topologically trivial deformations, the equigeneric locus $EG$ which parametrizes deformations of constant geometric genus, and the equiclassical locus $EC$ which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.)

It was previously known that the tangent space to $ES$ corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to $EG$ corresponds to the conductor ideal. We show that the support of the tangent cone to $EC$ corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of $ES$, $EC$, and $EG$. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.