Resolving singularities of curves with one toric morphism

Resolving Singularities of Plane Analytic Branches with One Toric Morphism

Let (C; 0) be an irreducible germ of complex plane curve. Let ? N be the semigroup associated to it and C ? C g+1 the corresponding monomial curve, where g is the number of Puiseux exponents of (C; 0). We show, using the specialization of (C;0) to (C ? ; 0), that the same toric morphisms

Resolving 3-dimensional toric singularities

This paper surveys, in the first place, some basic facts from the classification theory of normal complex singularities, including details for the low dimensions 2 and 3. Next, it describes how the toric singularities are located within the class of rational singularities, and recalls their main properties. Finally, it focuses, in particular, on a toric version of Reid’s desingularization strat...

One-Parameter Toric Deformations of Cyclic Quotient Singularities

In the case of two-dimensional cyclic quotient singularities, we classify all oneparameter toric deformations in terms of certain Minkowski decompositions introduced by Altmann [1]. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the ge...

One-parameter families containing three-dimensional toric Gorenstein singularities

(1.1) Let σ be a rational, polyhedral cone. It induces a (normal) affine toric variety Yσ which may have singularities. We would like to investigate its deformation theory. The vector space T 1 Y of infinitesimal deformations is multigraded, and its homogeneous pieces can be determined by combinatorial formulas developed in [Al 1]. If Yσ only has an isolated Gorenstein singularity, then we can ...

Resolving Singularities with Cartan’s Prolongation