## Programme

The aim of the workshop is to teach some interesting deformation theory to graduate students, postdoctoral fellows, and anybody else who wants to learn it. It's actually a school, with some talks by the young participants added. The workshop begins on Monday 10.00 a.m and ends on Friday 1.00 p.m. (schedule)

Call for abstracts. The participants that would like to contribute to the workshop with a talk (see schedule) should send title and abstract to the organizers.

Description of courses

Deformations of fibrations. The lectures will be devoted to the study of rational curves in the moduli space $\overline{M}_g$ of stable curves of genus $g$. This investigation will be reduced to the study of surfaces fibered over the projective line with fibres of genus $g$. The notion of ''free rational curve'' will be also studied with the purpose of obtaining informations on the uniruledness of $\overline{M}_g$ for some $g$.
REFERENCES:
R. Hartshorne: ''Deformation Theory'', Springer 2010.
E. Sernesi: ''Moduli of rational fibrations'' AG/0702865.
E. Sernesi: ''Deformations of Algebraic Schemes'', Springer 2006.

Derived deformation functors. For a deformation functor on Artinian rings, a derived deformation functor is an extension to differential graded (dg) or simplicial Artinian rings. The motivation for working with dg or simplicial algebras comes from various places, including obstruction theory and intersection theory. There are several formulations of derived deformation theory, and homotopy theory provides the tools to compare them.
REFERENCES:
The material covered will be a small subset of the following:
M. Kontsevich "Topics in algebra - deformation theory", available here,
V. Hinich "dg coalgebras as formal stacks"arXiv:math.AG/9812034,
M. Manetti "Extended deformation functors" arXiv:math.AG/9910071,
J.P. Pridham "Unifying derived deformation theories" arXiv:0705.0344,
and possibly parts of:
J.P. Pridham "Presenting higher stacks as simplicial schemes" arXiv:0905.4044.
I will try to use the bare minimum of homotopy theory, and none will be pre-requisite. A good starting point is the chapter on simplicial methods in:
C.A. Weibel "An introduction to homological algebra", CUP 1995.
For more detail or to fill in any gaps, see:
D. Quillen "Homotopical algebra", Springer LNM 1967.
M. Hovey "Model categories", AMS 1999.

Talks

MONDAY 15.25 - 16.10 :
Nathan Ilten: "Deformation theory of complete toric varieties".
Abstract: In this talk, I will present some results dealing with the deformation theory of toric varieties. I shall describe a combinatorial construction leading to one-parameter deformations of arbitrary toric varieties. In the smooth, complete case, these one-parameter families span the vector space of first-order deformations. Furthermore, it is easy to describe the relationship between the Picard groups of the fibers of such families for arbitrary toric varieties. Finally, I will discuss how this construction can be used to construct partial smoothings of certain projective toric varieties.

TUESDAY 15.25 - 16.10 :
Timo Schürg : "Deriving Deligne-Mumford Stacks with Perfect Obstruction Theories".
Abstract: In 1994 Kontsevich conjectured that moduli spaces having a 2-term complex controlling deformations and obstructions should be truncations of derived moduli spaces. I will present a proof of this conjecture in the category of derived stacks depeloped by Toen and Vezzosi.