TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 309, Number 2, October 1988 IDEALS ASSOCIATED TO DEFORMATIONS OF SINGULAR PLANE CURVES STEVEN DIAZ AND JOE HARRIS ABSTRACT. We consider in this paper the geometry of certain loci in defor- mation 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 con- stant 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. 1. Introduction. We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. Specificially, if p is a singular point of a reduced plane curve D, then we have an etale versal deformation of (D,p) (defined precisely in §3, see [Al]). pED ^ X i i 0 E B In the deformation space B we introduce three loci: (1) the equisingular locus ES C B which parametrizes equisingular deformations (the condition of equisingularity may be thought of as "topologically trivial"; a precise definition is given in §3); (2) the equigeneric or 6-constant locus EG C B which parametrizes deformations of (D,p) of constant geometric genus; (3) the equiclassical locus EC C B which parametrizes deformations of (D,p) of constant geometric genus and class (the class of a reduced plane curve is the degree of its dual). For example, if (D,p) is an ordinary cusp with equation D = {y2 — x3 = 0}, p = (0,0), then we may take for B the affine plane with coordinates a and b, and Received by the editors April 13, 1987 and, in revised form, July 22, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 14B07; Secondary 14H10, 14H20. The first author was almost totally supported by a National Science Foundation Mathemat- ical Sciences Postdoctoral Research Fellowship. The second author was partially supported by National Science Foundation grant number DMS-84-02209. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 433 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use