Simple {$K3$} singularities which are hypersurface sections of toric singularities

Toric embedded resolutions of quasi-ordinary hypersurface singularities

We build two toric embedded resolutions procedures of a reduced quasiordinary hypersurface singularity (S, 0) of dimension d . The first one provides an embedded resolution as hypersurface of (C, 0) as a composition of toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection (S, 0) → (C, 0) . This gives a positive answer to a a question of Lipm...

δm CONSTANT LOCUS OF VERSAL DEFORMATIONS OF NONDEGENERATE HYPERSURFACE SIMPLE K3 SINGULARITIES

Monodromy of Hypersurface Singularities

We describe algorithmic methods for the Gauss-Manin connection of an isolated hypersurface singularity based on the microlocal structure of the Brieskorn lattice. They lead to algorithms for computing invariants like the monodromy, the spectrum, the spectral pairs, and M. Saito’s matrices A0 and A1. These algorithms use a normal form algorithm for the Brieskorn lattice, standard basis methods f...

Topology of Hypersurface Singularities

Kähler’s paper Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle” offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. Kähler’s innovation of using a “square sphere” became standard in the toolkit of later researchers on singularities. We describe his contributi...

Minimal Discrepancies of Hypersurface Singularities

We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov’s conjecture is true for log-terminal threefolds.