Characterization of distinguished matrices of isolated hypersurface singularities through their spectral numbers
Sven Balnojan and Claus Hertling
Journal of Singularities
volume 30 (2026), 13-60
Received: 21 August 2025. In revised form: 13 March 2026
Abstract:
Isolated hypersurface singularities come equipped with distinguished bases of their Milnor lattices and with upper triangular integral matrices, which are called here distinguished matrices. These matrices form an orbit of a braid group and a sign change group. This paper proposes to characterize the distinguished matrices of singularities within all upper triangular integral matrices in terms of the variance of certain spectral numbers. It succeeds in the positive definite and the positive semidefinite cases. The ADE root lattices are crucial. In the semidefinite cases, results on nonreduced presentations of Weyl group elements are used.
2020 Mathematical Subject Classification:
32S25, 17B22, 06B15, 20F36, 20F55
Key words and phrases:
Isolated hypersurface singularities, upper triangular matrices, distinguished bases, spectral numbers
Author(s) information:
Sven Balnojan
email: svenbalnojan@gmail.com
Claus Hertling
Lehrstuhl für algebraische Geometrie
Universität Mannheim
B6 26
68159 Mannheim, Germany
email: hertling@math.uni-mannheim.de