TY - JOUR

T1 - Ideals associated to deformations of singular plane curves

AU - Diaz, Steven

AU - Harris, Joe

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1988/10

Y1 - 1988/10

N2 - 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.

AB - 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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U2 - 10.1090/S0002-9947-1988-0961600-2

DO - 10.1090/S0002-9947-1988-0961600-2

M3 - Article

AN - SCOPUS:0000472782

VL - 309

SP - 433

EP - 468

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -