A summer school in Catania

In the summer of 1997 we ran the PRAGMATIC summer school in Catania. After many different thoughts about the format, we decided to try an experiment: We made a list of 18 research problems (reproduced below) and handed it out on the first day of class. Early in the first week we went through the list briefly explaining the background of the problems, and asked each student to choose a problem to work on, either singly or with one or two others. We lectured on subjects necessary for understanding and attacking the problems: our lectures occupied each morning for four weeks. (There was a break of a fifth week in the middle in which the students worked on the problems without us.) In the afternoons in the first two weeks the students presented seminar talks on relevant papers, while in the last two weeks each student spoke on his/her own researches. In addition we met individually two or three times with the students working on each problem. (The local organizers had found us a lovely place to live by the Mediterranean sea, and most of the students were also able to join us for a mathematical lunch and a swim sometime during the program!) The choice of problems seemed to us very important. Our criteria were: that the answer was genuinely unknown and of interest to us; that some progress on the problem could be made by an industrious student using one of the computer algebra systems adapted to Algebraic Geometry, namely CoCoA [CoCoA], Macaulay [Bayer-Stillman], Macaulay2 [Grayson-Stillman], and Singular [Greuel-Pfister-Schönemann]; and that a strong student could, with luck and hard work, make a theoretical advance. We ran extensive tutorials and computer sessions to teach the students how to use Macaulay2. The students worked hard on their problems, and (it seemed to us) learned a good deal. In some cases they discovered results that were new and surprising to us. During the last week, the students were asked to write up what they had done. Most of these writeups appear in this volume. One student, whose project led to a correct proof for a well-known but unproved “theorem” of Severi, will appear elsewhere (see Russo [1998]). (Violo [1998] found a proof independently around the same time.) In the meantime, some other mathematicians have contributed to the solutions of some of the problems we posed; see the list of problems below for references of which we are aware. The Summer School culminated in a workshop run by the “Syzygies” node of Europroj. Several of the students had the excitement of presenting talks in the workshop on their