**
MINI-SCHOOL AT
THE BANACH CENTER
**

** "STRATIFICATIONS OF
MODULI SPACES"**

(partially supported by EAGER)

Invited speaker:

Gerard van der Geer (University of Amsterdam)

Place: Banach Center, Warsaw, Poland

Time: May 13. 2002 (arrival day) - May 19. 2002 (departure day)

Organizers: Grzegorz Banaszak and Piotr Pragacz

GENERAL PROGRAM: There will be five lectures by the invited
speaker and several "co-lectures" or additional lectures by participants.

SUMMARY OF LECTURES OF G. VAN DER GEER:

Stratifications on Moduli SpacesElliptic curves in positive characteristic come in two sorts:

ordinary and supersingular. This gives a stratification of the $j$-line

and a famous formula of Deuring gives the number of supersingular

elliptic curves in characteristic $p$. This can be generalized.

in two ways: curves of higher genus and abelian varieties of higher

dimension. We shall treat both generalizations. The $j$-line is

replaced by moduli spaces: the moduli spaces $M_g$ of curves of genus

$g$ and the moduli spaces $A_g$ of principally polarized abelian varieties

of dimension $g$. Stratifications can help us to understand the

cohomology and geometry of these moduli spaces. We shall begin by

recalling some properties of abelian varieties and

the construction of these moduli spaces. Our main emphasis

in the mini-course will be on the Ekedahl-Oort stratification on the

moduli of abelian varieties in postive characteristic, which generalizes

the stratification on the $j$-line defined by ordinary vs supersingular.

Some of the strata here correspond to well-known invariants of abelian

varieties in characteristic $p$, like the $p$-rank or the $a$-number.

in the mini-course will be on the Ekedahl-Oort stratification on the

moduli of abelian varieties in postive characteristic, which generalizes

the stratification on the $j$-line defined by ordinary vs supersingular.

Some of the strata here correspond to well-known invariants of abelian

varieties in characteristic $p$, like the $p$-rank or the $a$-number.

We shall treat local and global properties of strata of this

stratification and we determine their cycle classes in the

Chow ring. The formulas that we get can be seen as a generalization

of the formula of Deuring for the number of supersingular

elliptic curves. Applications of determination of the cycle classes

include results on the tautological rings and on complete subvarieties

of $A_g$. From this point of view moduli spaces in positive characteristic

seem more accessible than their characteristic zero counterparts.

But the characteristic $p$ results find applications in

characteristic zero too. We shall also discuss stratifications

on $M_g$, both in characteristic zero and $p>0$.

We treat related recent results on complete subvarieties.

BIBLIOGRAPHY SUGGESTED BY THE MAIN SPEAKER (most of these topics will be covered by co-talks
by the indicated co-speakers):

1)(G. Banaszak-P. Krason) elliptic curves; j-line; elliptic curves in positive
characteristic:

ordinary and supersingular; Frobenius, Verschiebung.

Literature: Silverman's book.

1a) A more specific topic, but a bit special, would be Deurings' mass

formula for the number of supersingular elliptic curves. My formulas

will specialize to this example, but a direct proof is possible and

might be amusing.

2) (G. Banaszak-P. Krason) Abelian varieties. Polarizations, Weil-pairing,
in positive characteristic:

Frobenius, Verschiebung, cohomology of an abelian variety (de Rham, l-adic).

Literature: Mumford's book; Birkenhake-Lange; manuscript of book in

preparation on http://www.science.uva.nl/~bmoonen

3) Moduli of abelian varieties; Siegel upper half plane,
symplectic group;

Satake compactification and maybe a bit a about toroidal compactifications.

It is very difficult to give good literature here.

Things get rather technical quickly , but if there is somebody who

knows the stuff and can give an accompanying non-technical talk

about compactifying A_g then that would be helpful.

4) (T. Szemberg) K3 surfaces; examples, cohomology of a K3 surface.

Again, it is difficult to give a unique source. Various books on surfaces

might help. If somebody could give a short survey on K3 surfaces,

giving examples, explaining their cohomology, the period map, mention

Torelli, then that would also help. If I am grossly underestimating

the audience (sorry for that!) then somebody could talk about

Deligne's result (see the paper in Lectures Notes 868 (page 58).

5) (P. Pragacz) A survey or introduction to the moduli of curves. Literature:

Mumford's Curves and their Jacobians, Harris, Morrison: Moduli of

curves. Alternatively, if I am underestimating the audience, then

somebody could give a talk on Diaz result that there is not a

complete subvariety of M_g of dimension g-1. (See page 288 of the

Harris-Morrison book.)

**A TENTATIVE DETAILED PROGRAM:**

TUESDAY (14.05.) 9:30-11:00 1st co-talk by G. Banaszak-P. Krason "On elliptic curves", 11:00-11:25 coffee/tee break, 11:25 Openning and Welcome, 11:30-12:30 1st talk by G. van der Geer, lunch break, informal discussions;

WEDNESDAY (15.05.) 9:30-11:00 2nd co-talk by G. Banaszak-P. Krason "On Abelian varieties", 11:00-11:30 coffee/tee break, 11:30-12:30 2nd talk by G. van der Geer, lunch break, 15:00-16:00 the talk by Y. Uetake , informal discussions;

THURSDAY (16.05.) **(only single talk in the morning!)** 9:30-11:00 3rd talk by G. van
der
Geer, 11:00-11:30 coffee/tee break, [12:15-14:00 Algebraic Geometry Seminar of A. Bialynicki-Birula
at Warsaw University], lunch break, 16:00-17:00 the talk by A.
Kuku; informal discussions;

FRIDAY (17.05.) 9:30-11:00 Co-talk by T. Szemberg "On K3 surfaces", 11:00-11:30 coffee/tee break, 11:30-12:30 4th talk by G. van der Geer, lunch break, 15:00 -16:00 Colloquium for Graduate Students by G. Banaszak, informal discussions;

SATURDAY (18.05.) 9:30-11:00 Co-talk by P. Pragacz "On moduli of curves", 11:00-11:30 coffee/tee break, 11:30 -12:30 5th talk by G. van der Geer (ending the school), lunch break, the afternoon sightseeing of Warsaw and its neighborhoods.

**ADDITIONAL TALKS:**

1. Yoichi UETAKE (AMU Poznan/Japan): Can one hear the sound of prime numbers?

2. Aderemi KUKU (ICTP Trieste/Nigeria): Profinite and continuous higher K-theory of exact categories, orders and groups.

3. Graduate Coloquium talk by Grzegorz BANASZAK (AMU Poznan): Algebraic K-theory and arithmetics.

Abstract: This lecture will discuss relations between algebraic K-theory of number fields and classical conjectures and problems in arithmetics.

All correspondence about the mini-school should be sent by e-mail to Grzegorz
Banaszak banaszak@amu.edu.pl or Piotr Pragacz pragacz@impan.gov.pl