The Italian school of algebraic geometry and Abel's legacy

THE ITALIAN SCHOOL OF ALGEBRAIC GEOMETRY AND ABEL’S LEGACY ALDO BRIGAGLIA, CIRO CILIBERTO, AND CLAUDIO PEDRINI Introduction Exactly one century ago, in the spring of 1902, the prestigious mathematical journal Acta Mathematica, decided to mark the centenary of Abel’s birth by dedicating three issues to the memory of this Norwegian mathematician. Some of the most influential Italian mathematician of the day were invited to contribute to these issues. Among those invited were Guido Catelnuovo, at the time the most well-known representative of the Italian school of algebraic geometry, and Federigo Enriques. Enriques, at first, accepted the invitation and, as he communicated to Castelnuovo, decided to write ... una specie di Bericht, breve, intorno ai nostri lavori, in cui mi propongo di esporre la teoria sotto l’aspetto analitico. ( ... a kind of Bericht, a short one, concerning our papers, in which I would like to expose the theory using the analytic viewpoint.) (letter of 13/3/1902, for all citations from the correspondence between Enriques and Castelnuovo, see [BCG]). The project was never completed. However this short citation is important for us: it tells us, in fact, that the Italian algebraic geometers, at the beginning of last century, used to look at their connections with Abel via analytic and transcendantal methods. Although never really central in the Italian school were, this attitude was always considered as a natural counterpart of the projective-geometric aspects typical of the Italians. Furthermore, as we will see, at the time Enriques was writing, there was in Italy an important revival of analytic ideas. In this paper we will try to make clear an interesting set of contributions which, inside the Italian school, were, in Enriques’ sense, inspired by Abel’s analytic viewpoint. For this we will go back to the real beginning of the italian school with Luigi Cremona, who is considered to be its founder. Then we will follow the evolution of these ideas up until the 1930’s and we will indicate how they are thought of today. Abel, apparently, never had any contact with the Italian mathematical environment,in his short life, nor even specifically during his travels to Italy (see [St]). Therefore, his influence on Italian mathematicians was late and indirect. However, we believe that, despite this, it has been deep and longstanding. In the first chapter we will give an overview of the influence of Abel’s ideas on the beginnings of the Italian school, filtered via Riemann’s viewpoint and the geometric interpretation of it by Clebsch and Gordan, and later by Brill and Noether. This chapter mainly centers around the character of Luigi Cremona, the founder of the Italian school. In the second chapter we will see how Abel’s influence was still active at the beginning of the XXth century and played a basic role in understanding the notion of irregularity of surfaces. The main characters, in this period, are the young Severi and the well established Caselnuovo, whose contributions we will review, indicating also their subsequent far reaching influence inside and outside the Italian school, until A. Weil’s proof of Riemann’s hypotheses for the zeta function of an algebraic curve over a finite field. The third chapter is devoted to Severi’s ideas on rational equivalence of 0-cycles on a surface. We will indicate how some of these ideas were related, in Severi’s mind, to Abel’s viewpoint. As is well known, Severi’s contributions on the subject have been very controversial. We will briefly report on the main criticisms but we will also try to elucidate which of them have a present validity. In particular we will direct our, and we hope the reader’s attention, to some of his ideas which are very closely related to Bloch’s conjecture. We dedicate a few, more technical, sections at the end of chapter 3 to open a window on the present developments of this last subject, specifically to some motivic interpretations which we think are rather close to Severi’s original viewpoint. Due to the different tastes and attitudes of the authors, which we deliberately did not make too much effort to hide, the paper is rather uneven. The first chapter is expository and more historiographical in nature. The other two, though sharing with the first a historiographical perspective, have a different flavour: in the second some technical aspects start to appear, and they become even more relevant in chapter 3, especially, as we said, in its last sections. We hope that the uneveness of the paper will attract, rather than repel, readers with different interests.