It has become customary for an author to acknowledge publicly his gratitude to those persons and institutions which have put him under an obligation, in various ways, during the period of preparation of his book. It is therefore an agreeable duty for me to record here that the greater part of my financial support, while plans for this volume were maturing and while it was being written, has come from the Rockefeller Foundation, through a grant made for that purpose to the New School for Social Research, and, since July 1944, from a fellowship awarded by the John Simon Guggenheim Memorial Foundation; to these institutions, and to all those persons who have been instrumental in obtaining for me such financial help,. I owe an enduring debt of gratitude. I owe no less, on the other hand, to a number of friends and colleagues (foremost among whom, if I may be allowed the pleasure of naming them here, have been C Chevalley, A Dresden and 0 Zariski) who have during that time, both in scientific and in personal matters, given me the benefit of their advice, suggestions and encouragement. My very cordial thanks go also to all those who read portions of the manuscript of this book, in preliminary or in final form, to whom many improvements are due, and to the Colloquium Committee and the staff of the American Mathematical Society, for doing me the honour of publishing this volume In their well-known series, and for the unfailing kindness and courtesy shown to me in all the arrangements connected- with this publication.
Sao Paulo, September 1, 1946
Algebraic geometry, in spite of its beauty and importance, has long been held in disrepute by many mathematicians as lacking proper foundations. The mathematician who first explores a promising new field is privileged to take a good deal for granted that a critical investigator would feel bound to justify step by step; at times when vast territories are being opened up, nothing could he more harmful to the progress of mathematics than a literal observance of strict standards of rigour. Nor should one forget, when discussing such subjects as algebraic geometry, and in particular the work of the Italian school, that the so-called "intuition" of earlier mathematicians, reckless in their use of it may sometimes appear to us, often rested on a most painstaking study of numerous special examples, from which they gained an insight not always found among modern exponents of the axiomatic creed. At the same time, it should always be remembered that it is the duty, as it is the business, of the mathematician to prove theorems, and that this duty can never be disregarded for long without fatal effects. The experience of many centuries has shown this to be a matter on which, whatever our tastes or tendencies, whether "creative" or "critical", we mathematicians dare not disagree. As in other kinds of war, so in this bloodless battle with an ever retreating foe which it is our good luck to be waging, it is possible for the advancing army to outrun its services of supply and incur disaster unless it waits for the quartermaster to perform his inglorious but, indispensable tasks. Thus for a time the indiscriminate use of divergent series threatened the whole of analysis; and who can say whether Abel and Cauchy acted more as "creative" or as "critical" mathematicians when they hurried to the rescue? One would be lacking in a sense of proportion, should one compare the present situation in algebraic geometry to that which these great men had to face; but there is no doubt that, in this field, the work of consolidation has so long been overdue that the delay is now seriously hampering progress in this and other branches of mathematics. To take only one instance, a personal one, this book has arisen from the necessity of giving a firm basis to Severi's theory of correspondences on algebraic curves, especially in the case of characteristic p ≠ 0 (in which there is no transcendental method to guarantee the correctness of the results obtained by algebraic means), this being required for the solution of a long outstanding problem, the proof of the Riemann hypothesis in function-fields. The need to remedy such defects has been widely felt for some time; and, during the last twenty years, various authors, among whom it will be enough to mention F Severi, B L van der Waerden, and more recently 0 Zariski, have made important contributions towards this end. To them the present book owes of course a great deal; nor is its title intended to suggest that further efforts in the same direction are now superfluous. No treatment of the. foundations of algebraic geometry may claim to be exhaustive unless it includes (among other topics) the definition and elementary properties of differential forms of the first and second kind, the so-called "principle of degeneration", and the method of formal power-series; but, concerning these subjects, nothing more than some cursory remarks in Chap. IX will be found in this book. Therefore some account of its exact scope, and -of its relationship to earlier work, must now be given.
The main purpose of the book is to present a detailed and connected treatment of the properties of intersection-multiplicities, which is to include all that is necessary and sufficient to legitimize the use made of these multiplicities in classical algebraic geometry, especially of the Italian school. At the same time, this book seeks to deserve its title by being entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals. in a treatment, of this kind, particular attention must be and has been given to the language and, the definitions. Of course every mathematician has a right to his own language - at the risk of not being understood; and the use sometimes made, of this right by our contemporaries almost suggests that the same fate is being prepared for mathematics as once befell, at Babel, another of man's great achievements. A choice between equivalent definitions is of small moment, and two theories which consist of the same theorems are to be regarded as identical, whatever their starting points. But in such a subject as algebraic geometry, where earlier authors left many terms incompletely defined, and were wont to make (sometimes implicitly) assumptions from which we wish to be free, all terms have to be defined anew, and to attach precise meanings to them is a task not unworthy of our most solicitous attention. Our chief object here must be to conserve and complete the edifice bequeathed to us by our predecessors. "From the Paradise created for us by Cantor, no one shall drive us forth" was the motto of Hilbert's work on the foundations of mathematics. Similarly, however grateful we algebraic geometers should be to the modem algebraic school for lending us temporary accommodation, makeshift constructions full of rings, ideals and valuations, in which some of us feel in constant danger of getting lost, our wish and aim must be to return at the earliest possible moment to the palaces which are ours by birthright, to consolidate shaky foundations, to provide roofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone. How much the present book contributes to this, our readers, and future algebraic geometers, must judge; at any rate, as has been hinted above, and as will be shown in detail in a forthcoming series of papers, its language and its results have already been applied to the re-statement and extension of the theory of correspondences on algebraic curves, and of the geometry on Abelian varieties, and have successfully stood that test.
Our results include all that is required for a rigorous treatment of so-called "enumerative geometry", thus providing a complete solution of Hilbert's fifteenth problem. They could be said, indeed, to belong to enumerative geometry, had it not become traditional to restrict the use of this phrase to a body of special problems, pertaining to the geometry of the projective spaces and of certain, rational varieties (spaces of straight lines, of conies, etc.), whereas we shall emphasize the geometry on an arbitrary variety, or at least on a variety without multiple points. The theory of intersection-multiplicities, however, occupies such a central position among the topics which constitute the foundations of algebraic geometry, that a complete treatment of it necessarily supplies the tools by which many other such topics can be dealt with. In deciding between alternative methods of proof for the theorems in this book, consistency, and the possibility of applying these methods to further problems, have been the main considerations. for instance, one will find here all that is needed for the proof of Bertini's theorems, for a detailed ideal-theoretic study (by geometric means) of the quotient-ring of a simple point, for the elementary part of the theory of linear series, and for a rigorous definition of the various concepts of equivalence. In consequence, the author has deliberately avoided a few short cuts; this is not to say that there may not be many more which he did not notice, and which our readers, it is hoped, may yet discover.
Our method of exposition will be dogmatic and unhistorical throughout, formal proofs, without references, being given at every step. A history of enumerative geometry could be a fascinating chapter in the general history of mathematics during the previous and present centuries, provided it brought to light the connections with related subjects, not merely with projective geometry, but with group-theory, the theory of Abelian functions, topology, etc.; this would require another book and a more competent writer. As for my debt to my immediate predecessors, it will be obvious to any moderately well informed reader that I have greatly profited from van der Waerden's well-known series of papers [published in the Math. Ann. between 1927 and 1938], where, among other results, the intersection-product has for the first time been defined (not locally, however, but only under conditions which ensure its existence "in the large"); from Severi's sketchy but suggestive treatment of the same subject, in his answer to van der Waerden's criticism of the work of the Italian school [in the Hamb. Abh., vol. 9 (1933), p. 335.]; and from the topological theory of intersections, as developed by Lefsehetz and other contemporary mathematicians. No direct use, however, will be made of their work; at the same time, I believe that whatever, in their results, pertains to the general intersection-theory on algebraic varieties is included in the theorems of the present volume, either as special cases or as immediate consequences. The attentive reader will also detect in many places the influence of 0 Zariski's recent work [In a number of papers published since 1940 in the Amer. J. Math., the Trans. Amer. Math. Soc., and the Ann. of Math. Detailed references need not be given here, especially since these investigations are soon to be published in book-form]; what he cannot easily imagine is how much benefit I have derived, during the whole period of preparation of this book, from personal contacts both with Zariski and with Chevalley, from their freely given advice and suggestions, and from access to their unpublished manuscripts.
Some brief details of the contents of the various chapters may now be given, more elaborate comments being reserved for Chap. IX. The first three chapters are preliminary, and intended to prepare the ground for the geometric theories which follow, by stating and proving all the purely algebraic results on which the latter depend. Chap. I and II are elementary, that is, they make no use of any result in abstract algebra beyond the general theory of abstract fields, and Hilbert's theorem of the existence of a finite basis for ideals of polynomials. The notion of specialization, the properties of which are the main subject of Chap. II, and (in a form adapted to our language and purposes) the theorem on the extension of a specialization (th. 6 of Chap. II, § 2) will of course be recognized as coming from van der Waerden. Chap. III is mainly devoted to the proof of the crucial theorem on the multiplicity of a proper specialization (th. 4 of Chap. III, § 4), on which our whole theory of intersection-multiplicities will rest. This is the only part of the book where "higher" methods of proof (viz. formal power-series, and the representation of an ideal in a Noetherian ring as intersection of primary ideals) are used; the reader who is willing to take that theorem for granted, or successful in constructing a simpler proof of it, will not require, in all the rest of the book, any knowledge of these methods, or of anything beyond what has been mentioned above. As will be indicated in Chap. IX, it is possible to prove the same theorem, by means of Zariski's results on birational correspondences, without making any use of formal power-series; on the other hand, Chevalley, by giving [Trans. Amer. Math. Soc., vol. 57 (1945), pp. 1-85], for some of the main results in the theory of intersections, alternative proofs which begin by establishing the corresponding theorems for algebroid varieties, has shown how the ring of formal power-series can be given the principal role, instead of the subordinate one which it plays in our treatment. Both authors make extensive use of the more technical parts of the abstract theory of ideals; this will be avoided in this book, by following a middle course (which is not, however, a compromise) between these two tendencies. It was not part of our purpose to investigate the connections between these several methods; this is a problem which still remains to be worked out.
The geometric language is then introduced in Chap. IV, which develops the elementary theory of algebraic varieties in affine spaces. The next two chapters contain the definition of intersection-multiplicities, which proceeds step by step, their main properties being stated and proved at every stage in such a way, that the next step can then be taken and these properties correspondingly extended. Chap. V deals with the intersections of an arbitrary variety and of a linear variety in an affine space, first (in § 1) when these varieties have complementary dimensions, then (in § 2) in general; § 3 contains some applications of these results to the theory of simple points. The general case is treated in Chap. VI, which includes all those results on intersection-multiplicities which are of a purely local nature.
A somewhat more explicit justification has to be given for the notions introduced in Chap. VII. It is well known that classical algebraic geometry does not usually deal with varieties in affine spaces, but with so-called projective models; the main feature which distinguishes the latter from the former is that they are, in a certain sense, "complete", or, in the topological case (when the ground-field is the field of complex numbers), compact. Nevertheless, local properties of varieties in projective spaces are almost always to be studied most conveniently on affine models of such varieties. There is now no reason why affine models, which can thus be pieced together so as to give a complete description of a given variety in a projective space, may not be pieced together differently; and there are problems (e.g. those concerning the Jacobian variety of a curve over a field of characteristic p ≠ 0, of which it is not known whether it possesses a one-to-one non-singular projective model) which cannot at present be handled otherwise than by such a procedure. This idea, inspired by the usual definition of a topological manifold by means of overlapping neighbourhoods, leads to the definition of an "abstract variety" in Chap. VII. The main definitions and results of the previous chapters, which are of a "local" nature, can be, extended without difficulty to such varieties; and new results "in the large" can be proved about them, because they can be assumed to be complete, while varieties in affine spaces can never be so. In particular, it is then possible to prove the theorem on intersections which provides the keystone for the whole theory; this is th. 8 of Chap. VII, § 5, a result closely related to the topological principle known as "Hopf's inverse homomorphism", and of the first importance, not only for the theory itself, but also for all its applications to specific geometric problems, since it enables one to introduce or withdraw at will m many auxiliary elements (points, varieties, etc.) as may be required at any time.
The last § of Chap. VII gives a translation of the main results of intersection-theory into a new language, particularly well adapted to applications, the "calculus of cycles". One main source of ambiguity, in the work of classical algebraic geometers and sometimes even in that of more modern writers, lies in their use of the word "variety", or of the word "curve" when they are dealing merely with the geometry on surfaces. As long as a "curve" or a "variety" is irreducible, there can be no uncertainty about it; but when, in the course of its "continuous variation" (however this may be defined), it splits up into several components, it is not always easy to know whether the resulting geometric entity is meant as a point-set, the union of irreducible varieties which need not even have the same dimension, or as a sum of varieties of the same dimension (a "virtual variety" in the sense of Severi), each multiplied with a well-determined integer which is its multiplicity. In the hope of doing away once for all with the resulting confusion (for those who will adopt our language, or at any rate an equivalent one which may be translated into ours term for term), two separate terms will be used here for these two kinds of entities, instead of the one term "reducible variety" which has previously been applied to both: "bunches of varieties" for the former, and (as in modem topology) "cycles" for the latter, while the word "variety" will be reserved for "absolutely irreducible algebraic varieties", i.e. for those which are irreducible over an algebraically closed ground-field and therefore remain so after an arbitrary extension of that field. An algebraic calculus of cycles can then be developed, closely analogous to the Algebra of homology-classes constructed by modern topologists: the main difference between the two is that, while the latter deals with classes, the former operates with the cycles themselves, but is unable, because of this, to have an intersection-product defined without any restrictive assumption. This, as will be seen, entails, in the practical handling of the calculus, a certain amount of inconvenience, which probably could be avoided, as the analogy suggests, by substituting, for the cycles, classes of cycles modulo a suitable concept of equivalence. If a coherent theory of linear equivalence could be built up, it would seem to be the best fitted for this purpose; at the present moment, "continuous" or even "numerical" equivalence would probably offer better prospects of immediate success. This book makes no attempt to proceed in that direction; but all the means for doing so are, it is hoped, provided in Chap. VII and Chap. IX.
Chap. VIII gives, on the basis of the results of the preceding chapters, a detailed treatment of the theory of divisors on a variety, both for its own sake and in order to provide the reader with some examples of the use of our calculus. The divisors which we consider here, and. which are defined by means of the intersection-theory, are substantially the same as the "divisors of the first kind" of modern algebraists; they are so at any rate on every normal variety, which, by Zariski's results (partly reproduced and extended in our Appendix II), is enough to make our theory applicable to all cases. The contents of this chapter include all that is needed for the theory of the linear equivalence of divisors (and, in particular, of "virtual curves", i.e., in our language, of cycles of dimension 1, on surfaces), and consequently for the foundation of the theory of linear series on a variety.
A point is thus reached in the systematic development of algebraic geometry, of which this volume may be regarded as the preliminary part, from which one may, with better perspective, look back on the course which has been hitherto followed, and make plans for the continuation of the voyage. This will be done in Chap. IX; it contains such general comments as could not appropriately be made before, formulates problems, some of them of considerable importance, and, in some cases, makes tentative suggestions about what seems to be at present the best approach to their solution; it is hoped that these may be helpful to the reader, to whom the author, having acted as his pilot until this point, heartily wishes Godspeed on his sailing away from the axiomatic shore, further and further into open sea.
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