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Let X be a quotient surface singularity, and define $G^{d e f} (X, r)$ as the directed graph of maximal Cohen-Macaulay (MCM) modules with edges corresponding to deformation incidences. We conjecture that the number of connected components of $G^{d e f} (X, r)$ is equal to the order of the divisor class group of X, and when X is a rational double point (RDP), we observe that this follows from a result of A. Ishii. We view this as an enrichment of the McKay correspondence. For a general quotient singularity X, we prove the conjecture...

The Denef-Loeser series for toric surface singularities.

Monique Lejeune-Jalabert, Ana J. Reguera (2003)

Revista Matemática Iberoamericana

Let H denote the set of formal ares going through a singular point of an algebraic variety V defined over an algebraically closed field k of charactcristic zcro. In the late sixties, J, Nash has observed that for any nonnegative integer s, the set js(H) of s-jets of ares in H is a constructible subset of some affine space. Recently (1999), J. Denef and F. Loeser have proved that the Poincaré series associated with the image of js(H) in some suitable localization of the Grothendieck ring of algebraic...

The discriminant of a real simple singularity

Eduard Looijenga (1978)

Compositio Mathematica

The Discriminants of a Series of Unimodular Singularities of Space Curves.

Klaus Wirthmüller (1987)

Mathematische Annalen

The duality of cusp singularities.

Masa-Nori Ishida (1992)

Mathematische Annalen

The embedding dimension of the formal Moduli space of certain curve singularities.

Bernd Ulrich (1982)

Manuscripta mathematica

The end curve theorem for normal complex surface singularities

Walter D. Neumann, Jonathan Wahl (2010)

Journal of the European Mathematical Society

We prove the “End Curve Theorem,” which states that a normal surface singularity $(X, o)$ with rational homology sphere link $Σ$ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function $(X, o) \to (ℂ, 0)$ whose zero set intersects $Σ$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” $(X, o)$ is described by giving an explicit set of equations describing...

The Euler characteristic of algebraic complete intersections.

Zbigniew Szafraniec (1989)

Journal für die reine und angewandte Mathematik

The Formal Hodge Filtration.

Arthur Ogus (1976)

Inventiones mathematicae

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