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

Yongnam Lee, *Sogang University*

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$Q$-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.

#### References

- 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