Summary: PResolutions of Cyclic Quotients from the Toric Viewpoint
Institut f¨ur reine Mathematik, HumboldtUniversit¨at zu Berlin
Ziegelstraße 13A, D10099 Berlin, Germany.
(1.1) The break through in deformation theory of (twodimensional) quotient singularities
Y was Koll'ar/ShepherdBarron's discovery of the onetoone correspondence between socalled P
resolutions, on the one hand, and components of the versal base space, on the other (cf. [KS],
Theorem (3.9)). It generalizes the fact that all deformations admitting a simultaneous (RDP)
resolution form one single component, the Artin component.
According to definition (3.8) in [KS], Presolutions are partial resolutions ß : ~
Y ! Y such that
ffl the canonical divisor K ~
Y jY is ample relative to ß (a minimality condition) and
Y contains only mild singularities of a certain type (socalled Tsingularities).
Despite their definition as those quotient singularities admitting a I
smoothing ([KS], (3.7)), there are at least three further descriptions of the class of Tsingularities: