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Transactions of the American Mathematical Society

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Ideals associated to deformations of singular plane curves

Authors: Steven Diaz and Joe Harris
Journal: Trans. Amer. Math. Soc. 309 (1988), 433-468
MSC: Primary 14B07; Secondary 14H20
MathSciNet review: 961600
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Abstract: 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.

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