** **

In the period **7 June
- 8 July 2000**, at the

Mathematics Department of the **University of Catania**,

a big edition of this school of research **Pragmatic 2000** will be held.

Namely, for this edition we will have two distinct sections:

First period: **June 7 - June 23, 2000**

**Teachers**: A.V. Geramita and Juan
Migliore (with the collaboration of A. Bigatti and C. Peterson)

**Themes of research**:

By its nature, algebraic geometry studies the interplay between both algebraic and geometric concepts. In the roughly 20 hours of this course we would like to take some examples of kinds of commutative rings and kinds of geometric ideas and consider the rings from a geometric aspect and the geometric ideas from an algebraic one. In this brief outline of the course we shall only mention topics that fall into the scope of the things we have in mind. With only 20 hours available some of these topics may, in the final analysis, have to be omitted or simply left to the student for independent study (with guides to the literature given by us).

COHEN-MACAULAY RINGS:

algebraic definition, preservation under certain algebraic operations,
relation to completeness of linear systems (Zariski's Theorem, Serre's Theorem),
saturation of hyperplane sections of algebraic varieties, graded Betti numbers
of varieties and their hyperplane sections, the "Castelnuovo method", classification
of arithmetically Cohen-Macaulay varieties of low codimension (Hilbert-Burch
Theorem), extremal Cohen-Macaulay varieties. Among the C-M varieties are the
finite sets of points in **P**^n and connected to them we will discuss:
conjectures about the ideal of a finite set of points in **P**^n, using
points (blowing-up) to construct embedded rational n-folds, using the ideal
of the points to explore the finite free resolution of the defining ideals
of these embedded n-folds, connections with the Rees Algebra of ideals connected
to these sets of points.

GORENSTEIN RINGS:

complete intersections and liaison, subcanoncial curves, Gorenstein
linkage, Hilbert functions of Gorenstein rings, the Cayley-Bacharach property,
construction of Gorenstein varieties (points, curves, codimension 2 and three,
higher codimension), Gorenstein varieties with given Hilbert function (and
given resolution), classifying spaces for Gorenstein Artinian rings and catalecticant
varieties, the strange Gorenstein rings of Bernstein-Iarrobino and Boij-Laksov,
Gorenstein Rings with the Weak Lefschetz Property.

BUCHSBAUM RINGS:

algebraic definitions, relation to liaison, construction of
Buchsbaum varieties, Buchsbaum curves and Buchsbaum varieties of higher dimension.

Second period: **June 26 - July 8, 2000**

**Teachers**: K. Hulek and K. Ranestad

**Themes of research**:

ABELIAN VARIETIES: Moduli spaces of abelian varieties are much studied objects in algebraic geometry and arithmetic geometry. Mumford et al. introduced the concept of toroidal compactifications of these moduli varieties. Recently much progress has been achieved in giving specific toroidal compactification, namely the second Voronoi compactification, an interpretation as a moduli space, i.e. the boundary points correspond to certain degenerate abelian varieties in a functional way. This progress is due to Alexeev and Nakamura and is based on a better understanding of Munford's construction of degenerations of abelian varieties, thus continuing works of Faltings and Chai. The idea of the course is to introduce the audience to this technically sometimes demanding area by working out a number of specific examples and cases of small polarization.

VARIETIES OF SUMS OF POWERS: The interrelation
between the topics is the example of the moduli space of (1,7)-polarized abelian
surfaces that is isomorphic to the variety of sums of powers of the Klein
curve. On the second topic we think of explaining apolarity and use it to
investigate varieties of various canonical forms (sums of powers is an example),
as well as the properties of the dual socle polynomial of the intersection
ring of various varieties. We shall spend most of the time on examples.

**Pragmatic** will grant about 10
**fellowships** which will permit the fellowship holders to attend the
above lectures and seminars and will pay the full cost of board and lodging
in Catania for the entire period of the school and will also cover some travel
expenses.

The applicants should choose one of the
two sessions for which they intend to apply. According to the number of requests
the Committee of **Pragmatic** may permit some participants to follow both
sessions. These fellowships will be awarded to young European researchers,
say at the pre-postdoctoral level, with preference given to those belonging
to geographically isolated locations.\medskip In addition the Committee of
**Pragmatic** will admit to the Courses and Seminars some other people
who can be supported by their own Institutions. The Pragmatic Committee will
offer help in finding suitable accomodations at a reasonable cost for these
participants.

The Co-ordinator of **Pragmatic**
is Prof. Alfio Ragusa (University of Catania). Other local organizers are:

Rosario Strano, Renato Maggioni, Giuseppe
Paxia, Salvatore Giuffrida.

People who wish intend to be considered
either for a fellowship or for admission to the courses should fill out the
enclosed **application form**, which should then be sent to **Prof.Alfio
Ragusa** at the following address:

E-mail: Ragusa@dipmat.unict.it

or by ordinary mail to: Prof. Alfio Ragusa-Dipartimento di Matematica- Viale A.Doria, 6 Catania-95125-Italy.

A fee of 150 EURO (payable at registration)
will be requested from all participants for each session (**the fellowship
will cover the cost of this fee**).

**The deadline** for applications
is December 31, 1999.

The Committee of Pragmatic will decide about fellowships and admissions by January 31, 2000 and will inform all the applicants of the outcome.

Moreover a list of participants and all
the activities of the school will be spread by **EAGER** net.

Interested people who need more information
about **Pragmatic** can contact any member of the local committee, using
the following E-mail address:

Strano, Paxia, Maggioni, Giuffrida (one of these)@dipmat.unict.it