COCOA VI COmputational COmmutative Algebra and International School on Computer Algebra Villa Gualino, Torino, Italy May 31 - June 5, 1999

Dear Students,

The course "Finite Sets of Points" will begin with four lectures by Robbiano on

1. basics of Gröbner Bases (see [KR] below);
2. basics of Hilbert Functions and Hilbert-Poincare' Series;
3. Points from Combinatorics: lifting of monomial ideals;
4. Points from Algebra, Interpolation, Statistics;
5. Hyperplane sections and points in projective spaces.

For b, c, d, e notes will be given at the school.

The remaining four lectures of this course will be given by Geramita, who will concentrate on some problems with a more geometric flavour (naturally with a strong computational component). These talks will center around a collection of open problems regarding the Hilbert functions and resolutions of ideals of points in projective space. The conjectures discussed will be:

1. The Ideal Generation Conjecture for generic points in P^n (and its artinian analogue).
2. The Minimal Resolution Conjecture (and the point counter-examples of Eisenbud-Popescu) and its artinian analogue (with no counterexamples known).
3. Ranges of resolutions permitted for a given Hilbert function of points.
4. The classification of Hilbert functions of points with the Uniform Position Property,

• Hilbert functions of Fat Points and Waring's Problem for Forms.

For some good background reading (which will be useful for all the talks given in the school) we recommend the draft of the new book

   [KR] Computational Commutative Algebra    by    Kreuzer and Robbiano

As far as the material in the section by Geramita is concerned, the following notes and papers will be useful (other items are listed on Geramita's web page)

  [GGR]  Geramita, A. V., Gregory, D., Roberts, L., 
         Monomial Ideals and Points in Projective Space, 
         Jo. of Pure & Applied Alg., Vol. 40, (1986), pp. 33-62. 

  [GM]   Geramita, A. V., Migliore, J. C., 
         Hyperplane Sections of Curves in P3. 
         Communications in Algebra, Vol. 17 (12) (1989), 3129-3164. 

  [BGM]  Geramita, A. V., Bigatti, A., Migliore, J. C., 
         Geometric Consequences of Extremal Behaviour 
         in a Theorem of Macaulay, 
         T.A.M.S., Vol. 346, 1994, 203-235. 

  [DG]   Diaz, S., Geramita, A. V., 
         Points in Projective Space in Very Uniform Position, 
         Rend. Sem. Math. Univers. Politech. Torino, Vol. 49, 2,
	 (1991), 267-280. 

  [G]    Geramita, A. V., 
         Zero-Dimensional Schemes: Singular Curves and Rational Surfaces,
         Proc. of Int. Conf. on Zero-dimensional Schemes 
         (Ravello, Italy, June 1992), W. de Gruyter, 1994. 

Among the OTHER PUBLICATIONS (i.e. category c) ) the expository articles 12) and 13)i) would probably be most useful, i.e.

  The Curves Seminar at Queen's, Vol. IX (1993), 
  edited by A. V. Geramita, Queen's Papers in Pure and Applied
  Mathematics, No. 95 (+7 lectures therein). 

  The Curves Seminar at Queen's, Vol. X (1996), 
  edited by A. V. Geramita, Queen's Papers in Pure and Applied
  Mathematics, No. 102 (and 2 articles therein).  
     i Inverse Systems of Fat Points: Waring's Problem, Secant
       Varieties of Veronese Varieties and Parameter Spaces for
       Gorenstein Ideals, pgs.3-104.  

We are very much looking forward to seeing you in a few weeks.

With warm regards, Tony Geramita and Lorenzo Robbiano