Research school in
Sponsored by PRIN 2011, PRA UNICT, GNSAGA
Moduli of Curves and Line Bundles
Catania, Italy, June 22nd - July 10th, 2015
Prof. Alessandro Chiodo
Prof. Filippo Viviani
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.
June 22nd - July 10th, 2015
(Arrival date June 21st; departure date July 11th)
Prof. Alessandro Chiodo - Université de Paris 6
Prof. Filippo Viviani - Università Roma Tre
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
Calabi-Yau varieties (X,Y) of dimension three
which are dual in the sense of mirror symmetry: hp,q(X;C)
= h3-p,q(Y; C).
We will formulate a more general statement and a proof which will allow
us to present a new approach (Fan-Jarvis-Ruan) to the quantum
cohomology of Calabi-Yau varieties by means of r-spin curves. These
are algebraic curves of genus g equipped with a line bundle L, whose rth power is isomorphic to the canonical bundle: Lr ≅ ƒÖ.
The geometry of the moduli spaces of r-spin
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
(Polishchuk-Vaintrob) has recently allowed a new approach to the
problem of computing virtual intersection numbers.
The moduli space Mg of smooth and projective curves of genus g and the moduli space Ag of principally polarized abelian varieties of dimension g are certainly among the most studied moduli spaces. The Torelli morphism from Mg to Ag, sending a curve into its Jacobian, establish a deep bridge between the two moduli spaces.
The moduli spaces Mg and Ag admit compactifications compatible with the Torelli morphism. On one hand, Deligne-Mumford introduced a modular compactification of Mg parametrizing stable curves. On the other hand, Ash-Mumford-Rapoport-Tai introduced several toroidal compactifications of Ag, 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 Ag, parametrizing stable semiabelic pairs. The Torelli morphism extends from the Deligne-Mumford compactification of Mg to some of the toroidal compactifications of Ag and also to the Alexeev modular compactification, by sending a stable curve into its canonical compactified Jacobian in degree g-1.
More recently, tropical analogues of the moduli spaces Mg and Ag 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 non-archimedean 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 Mg and Ag 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 Mg and Ag, the tautological rings of Mg and Ag, compactified Jacobians of singular curves, the compactified universal Jacobian and its birational geometry.