Summer Lectures in Geometry   RSS

23/07/2012, 15:45 — 16:45 — Room P3.10, Mathematics Building
, Sogang University

Q-Gorenstein deformation theory and its applications

The 1st lecture will review the singularity of class T and Q-Gorenstein deformation theory. The notion of singularity of class T, which is defined as a quotient surface singularity admitting a Q-Gorenstein smoothing, was introduced by Kollár and Shepherd-Barron. They also gave an explicit description of the singularity of class T. The notion of Q-Gorenstein deformation is popular in the study of degenerations of normal algebraic varieties in characteristic zero related to the minimal model theory and the moduli theory since the paper by Kollár and Shepherd-Barron. A typical example of Q-Gorenstein deformation appears as a deformation of the weighted projective plane P(1,1,4): Its versal deformation space has two irreducible components, in which the one-dimensional component corresponds to the Q-Gorenstein deformation and its general fibers are projective planes. By developing the theory of Q-Gorenstein deformation functor, we can generalize their results to surfaces in positive characteristics.


  • J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299-338.
  • Y. Lee and J. Park, A simply connected surface of general type with p g=0 and K 2=2, Invent. Math. 170 (2007), 483-505.
  • Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, preprint 2011 (arXiv:1103.5185, to appear in PLMS).
Session 1

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