Foundations of Algebraic Geometry

Abstract

ALGEBRAIC geometry, in spite of its beauty and importance, has long been reproached for lacking proper foundations. Great discoveries have been made, especially in Italy, by the intuition of a number of gifted workers, whose insight had been cultivated by a careful study of special cases. But the time for placing the subject on a firm foundation is long overdue, and the delay is seriously hampering progress. For the last twenty years a few geometers, such as Severi, van der Waerden and Zariski, have made important contributions towards this end. Now Prof. Weil has given a connected and comprehensive account of his own and other researches, sufficient to justify the use of intersection-multiplicities, which play such a large part in recent geometry, especially in Italian work. The book is not easy reading, especially for those to whom geometry is synonymous with space-intuition. At a first glance the reader might imagine that it is a treatise on something like abstract algebra. Indeed, the author warns the reader that it is necessary to have a good elementary knowledge of abstract fields and of other topics that will be found in van der Waerden‘s Moderne Algebra. Apart from this the book is self-contained, and its subject is not abstract algebra, but what might be called abstract geometry. Foundations of Algebraic Geometry By Prof. Andre Weil. (American Mathematical Society: Colloquium Publications, Vol. 29.) Pp. xix+288. (New York: American Mathematical Society, 1946.) 5.50 dollars.