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Abstract des Vortrages von Jesse Leo Kass

How to deform a n-by-(n+1) matrix

At the 1974 ICM E. Brieskorn delivered a lecture by V. I. Arnold in which he explained a program for studying a singularity defined by one equation --- a hypersurface singularity --- by deforming the equation, and over the past 39 years remarkable progress has been made. One of the great achievements was the discovery that the simple hypersurface singularities are exactly the ADE singularities and many features of these singularities, such as the Milnor lattice, can be described by the ADE Dynkin diagrams. What about more general classes of singularities? A natural class to consider is the codimension 2 singularities. Such a singularity can be given by equations that are the minors of a n-by-(n+1) matrix by a celebrated theorem of Burch/Hilbert/Schaps, and the singularity can be studied by deforming the matrix. Frühbis-Krüger and Neumer classified these singularities, and in important special cases the Milnor lattice of a simple codimension 2 singularity was computed by Alpert, Ebeling, Ebeling/Guseĭn-Zade, Mond/van Straten, and Tyurina.

In my talk I will present a uniform technique for computing the Milnor lattice of a simple codimension 2 singularity or more generally a codimension 2 singularity of Cohen-Macaulay type 2. This technique establishes an unexpected connection between hypersurface singularities and codimension 2 singularities, and suggest the importance of studying deformations of a nonreduced curve singularity.