Deformations of plane curve singularities of constant class
Date of Award
Doctor of Philosophy (PhD)
Steven P. Diaz
Plane curve, Singularities, Constant class
This dissertation considers the geometry of the locus of constant class in the deformation spaces of plane curve singularities. In [DH], Diaz and Harris discuss the geometry of the equisingular ( ES ), equigeneric ( EG ), and equiclassical ( EC ) loci in the same deformation spaces. We define the locus of constant class EK as the locus which parametrizes deformations of constant class. By definition, EK contains EC . We investigate and answer the question: Is EK equal to EC ?
We define conditions for EK to be different from EC and then explore the singularities where this might be possible. For y 2 + x n = 0, we are able to show where EK is different from EC . We also compute the tangent cones for the different pieces of EK and hence for EK itself, in many cases. Investigating the y 3 + x n = 0 singularities in search of the extra pieces of EK leads to the definition of the pre-EK loci , each of which possibly contains a piece of EK and other loci. We explore the possibilities for these other loci, finally leading up to the double triple point locus in one of the pre-EK loci for y 3 + x 6 = 0.
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Lynn, Philip Joseph, "Deformations of plane curve singularities of constant class" (2006). Mathematics - Dissertations. Paper 22.