Ideals associated to deformations of singular plane curves. (English) Zbl 0707.14022
Summary: 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.
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.
MSC:
| 14H20 | Singularities of curves, local rings |
| 14H10 | Families, moduli of curves (algebraic) |
| 14D15 | Formal methods and deformations in algebraic geometry |
| 14B07 | Deformations of singularities |
| 14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |
Keywords:
deformation spaces of plane curve singularities; equisingular locus; equigeneric locus; equiclassical locus; dimensions of families of plane curvesReferences:
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