"EAGER is a European Algebraic Geometry Research Training Network
supported by the HPP programme of the European Commission"
 

Eager Advanced School in Algebraic Geometry

Eager

Centro Internazionale per la Ricerca Matematica, CIRM (Trento)

Dipartimento di Matematica, Politecnico di Milano

Levico Terme, Trento, Italy

September, 1-5, 2001

Moduli spaces of curves

Second Announcement

"This school is sponsored by the HPP programme of the European Commission
as a high-lever scientific conference (see grants below)."


Lecturers: Lucia Caporaso, Joe Harris, Ian Morrison

Program

Each of the three courses will consist of five one hour lectures. The programs described below will be covered in four of the lectures. The additional lecture in each course will be devoted to filling in necessary preliminaries (based on discussions with the participants) or to new results, research topics, open problems.

Lucia Caporaso will describe some of the basic moduli spaces and apply the theory to study families of curves and their arithmetic properties.

1. and 2. Construction (using Geometric Invariant Theory) of the moduli space of stable curves, the compactified universal Jacobian over it, moduli spaces for vector bundles over stable curves. Picard groups of the moduli space of curves and of the universal curve over it.

3. Families of curves. Positivity properties of the relative dualizing sheaf of a family of curves. Application: the uniform Parshin-Arakelov Theorem and the Correlation Theorem on fibered powers of families of curves.

4. Applications to arithmetic.

Uniformity of rational points of curves over number field and over function field.

Joe Harris's lectures will focus on plane curves, illustrating the rich interplay between the moduli theory of abstract curves and the geometry of concrete curves in projective space.

1. Plane curves and their parameter spaces.

Introduction of Severi varieties and basic properties. Discussion of central questions: dimension, local geometry, irreducibility, degree. Related topics: the Enriques'conjecture, generalizations of Severi varieties.

2. Deformation theory and its applications to dimension counts and local geometry. Plane curves with non-ordinary singularities and with tangency conditions. Semistable reduction and its use to describe flat limits of plane curves.

3. Proof of the irreducibility Theorem: the approaches of Severi and Zarisky and reduction of the statement to exhibition of degenerations. Application of semistable reduction to prove the theorem.

4. Degrees of Severi varieties and enumerative issues. Reducible hyperplane sections of Severi varieties, and the recursive formula for their degrees.

Ian Morrison will lecture on the birational geometry of the moduli spaces of pointed stable curves.

1. Nef and ample divisor classes on the moduli space of stable curves. Admissible covers and divisors on the hyperelliptic locus. Proof of the Cornalba-Harris theorem.

2. Kodaira dimension of Mg. Sketch of the Eisenbud-Harris proof of the Harris-Mumford Theorem.

3. Mori and nef cones of moduli spaces of pointed stable curves: conjectural descriptions and various reductions of the conjectures.

4. Applications to semi-ampleness and questions about existence and structure of fibrations and contractions. Related results on birational maps.

Schedule

Each speaker will give a morning lecture daily. Afternoons will be scheduled with the primary aim of giving the participants time to digest the morning lectures through independent work, informal discussions and consultations with the speakers. In addition, a few junior lectures per day will be scheduled each afternoon starting at 17.00. The morning lecture schedule is:

Caporaso 9:00-10:00

Harris 10:30-11:30

Morrison: 12:00-13:00

General information

The school will take place in Levico, a resort near Trento (Italy), at the Grand Hotel Bellavista.
The arrival of participants will be on Friday, August 31. Classes will be from Saturday, Sept. 1, 2001, to Wednesday, Sept. 5, 2001. The departure of participants will be on Wednesday afternoon.

There is a registration fee of 150.000 ITL, to be paid cash at the arrival. The full board accomodation in double room will cost 95.000 ITL/day.

Updated information, maps and schedules for reaching Levico Terme can be found at the web page

http://www.science.unitn.it/cirm/travelbella.html

Other interesting web pages concerning the region

http://www.unitn.it/trentino/trentino.html

http://www.valsugana.nu

Grants

Several grants for young European researchers are available (covering accomodation and/or travel expenses). We particularly encourage young women researchers and young researchers from less
favoured regions of the European Commission to apply for support. If you want support please fill the
form below and send it before June, 30.

Reply

If you are interested in participating, please send the form(s), preferably by e-mail, to

Augusto Micheletti, Secretary of CIRM
Istituto Trentino di Cultura
I- 38050 POVO (trento), Italy
fax: Italy+0461 810629

e-mail: michelet@science.unitn.it

or contact the local organizer

Giorgio Bolondi
Dipartimento di Matematica
Politecnico di Milano
giobol@mate.polimi.it
 

Registration Form

Application Form for Financial Support