Classification of Unimodal Parametric Plane Curve Singularities in Positive Characteristic

Abstract

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove.

On the Computation of the Codimension of Map Germs Using the Lie Algebra Associated with a Restricted Left–Right Group

“…In the history of the theory of singularities of map germs, A-equivalence has been the most natural equivalence among map germs from the view point of differential topology. Group A, the tangent space to the orbit under the action of this group and its codimension play an important role in the classification of map germs (see [1][2][3][4][5][6][7][8][9][10][11][12]). In [13], the authors gave an algorithm to compute the codimension of map germs under an A-equivalence.…”