Research school in
Algebraic Geometry and Commutative Algebra

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

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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)


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 ≅ . 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.

Moduli Spaces of Curves and Abelian Varieties (Prof. F. Viviani)

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.

Local committee:

Alfio Ragusa
Francesco Russo
Giuseppe Zappalà

People who wish to be considered either for participation and/or for financial support should fill out the enclosed application form, which should be sent to one of the following addresses:

The deadline for applications is March 15th, 2015. The committee of Pragmatic will decide about financial supports and admissions within March 31st, 2015. Interested people who need more information about Pragmatic can contact any member of the local committee.

Pragmatic 2015 will cover the boarding and lodging expenses of the participants which are not supported by their home institutions. Travel expenses are not funded.

Application form to Pragmatic 2015



Male or Female:

Date of birth:



Academic qualification (PhD, Master,...):

Name of advisor:

Professional status and current position:

E-mail address:

Field of interest:

Are you supported by your home institution? (Possible answers: totally, partly, no)

If you have a PhD, when did you get it and where?

Reasons for wanting to participate in the session:

It is suggested that the applications include both a recommendation letter and a brief curriculum vitæ