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PResolutions of Cyclic Quotients from the Toric Viewpoint Klaus Altmann
 

Summary: P­Resolutions of Cyclic Quotients from the Toric Viewpoint
Klaus Altmann
Institut f¨ur reine Mathematik, Humboldt­Universit¨at zu Berlin
Ziegelstraße 13A, D­10099 Berlin, Germany.
E­mail: altmann@mathematik.hu­berlin.de
1 Introduction
(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
ffl ~
Y contains only mild singularities of a certain type (so­called T­singularities).
Despite their definition as those quotient singularities admitting a I
Q­Gorenstein one­parameter
smoothing ([KS], (3.7)), there are at least three further descriptions of the class of T­singularities:

  

Source: Altmann, Klaus - Fachbereich Mathematik und Informatik & Institut für Mathematik, Freie Universität Berlin

 

Collections: Mathematics