PRAGMATIC 2015

Research school in Sponsored by PRIN 2011, PRA UNICT, GNSAGA 
Moduli of Curves and Line Bundles
Catania, Italy, June 22nd  July 10th, 2015 Lecturers: Prof. Alessandro Chiodo Prof. Filippo Viviani 
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1997 
Pragmatic 2015
(First announcement) P.R.A.G.MAT.I.C. is a project, sponsored by PRIN 2011, PRA UNICT, and GNSAGA, for stimulating research in algebraic geometry and commutative algebra among young people, especially those living in isolated centres or peripheral universities all over Europe. For this purpose, every year, a group of experts suggests a set of problems in some specific field and gives a series of introductory lectures. The participants have a chance to choose a problem and pursue its solution in small groups and in consultation with the experts. At the end of all the activities of Pragmatic the participants have the opportunity to submit their result to a special issue of the journal Le Matematiche.
This year's event of Pragmatic will be held in June 22nd  July 10th, 2015,
at the Dipartimento di Matematica e Informatica
of the Università of Catania.
Catania, Italy
June 22nd  July 10th, 2015 (Arrival date June 21st; departure date July 11th) Lecturers: Prof. Alessandro Chiodo  Université de Paris 6 Prof. Filippo Viviani  Università Roma Tre Collaborators: Giulio Codogni  Università Roma Tre Jérémy Guéré  Université de Paris 6 Topic: Moduli of Curves and Line Bundles The goal of the school is to introduce the participants to the theory of moduli of curves with particular attention to problems related to the enumerative geometry of curves and line bundles. This subject has been at the center of new research areas with many accessible open problems, particularly for participants who are willing to work in quantum cohomology and/or birational geometry of moduli spaces. There will be two introductory courses. Spin Curves and Mirror Symmetry (Prof. A. Chiodo)
We will start from explicit examples of a phenomenon which is
apparently completely unrelated to the subject of the event: pairs of
CalabiYau varieties (X,Y) of dimension three
which are dual in the sense of mirror symmetry: h^{p,q}(X;C)
= h^{3p,q}(Y; C).
We will formulate a more general statement and a proof which will allow
us to present a new approach (FanJarvisRuan) to the quantum
cohomology of CalabiYau varieties by means of rspin curves. These
are algebraic curves of genus g equipped with a line bundle L, whose rth power is isomorphic to the canonical bundle: L^{r} ≅ ƒÖ.
The geometry of the moduli spaces of rspin
curves is still largely unexplored and recent techniques give access to
many open problems. These can be intrinsically related to the geometry
of the moduli space or oriented towards quantum cohomology and mirror
symmetry, where a construction in terms of matrix factorisations
(PolishchukVaintrob) has recently allowed a new approach to the
problem of computing virtual intersection numbers.
The moduli space M_{g} of smooth and projective curves of genus g and the moduli space A_{g} of principally polarized abelian varieties of dimension g are certainly among the most studied moduli spaces. The Torelli morphism from M_{g} to A_{g}, sending a curve into its Jacobian, establish a deep bridge between the two moduli spaces. The moduli spaces M_{g} and A_{g} admit compactifications compatible with the Torelli morphism. On one hand, DeligneMumford introduced a modular compactification of M_{g} parametrizing stable curves. On the other hand, AshMumfordRapoportTai introduced several toroidal compactifications of A_{g}, depending on the choice of a polyhedral decomposition of the cone of positive definite quadratic forms. More recently Alexeev (and subsequently Nakamura and Olsson) introduced modular compactifications of A_{g}, parametrizing stable semiabelic pairs. The Torelli morphism extends from the DeligneMumford compactification of M_{g} to some of the toroidal compactifications of A_{g} and also to the Alexeev modular compactification, by sending a stable curve into its canonical compactified Jacobian in degree g1. More recently, tropical analogues of the moduli spaces M_{g} and A_{g} have been introduced, as well as a tropical analogue of the Torelli morphism. These tropical moduli spaces can be seen as canonical skeleta of the nonarchimedean Berkovich analytifications of the classical moduli spaces.
The aim of this course is to give an introduction to the beautiful geometry underlying the moduli spaces M_{g} and A_{g} and the Torelli morphism, starting from the classical case and then moving to the compactifications and tropicalizations.
Moreover, we will touch upon related topics, like: the birational geometry of M_{g} and A_{g}, the tautological rings of M_{g} and A_{g}, compactified Jacobians of singular curves, the compactified universal Jacobian and its birational geometry.
Alfio Ragusa pragmatic@dmi.unict.it 