For several beautiful and expert discussions of the contrasts and relations between classical and the evolving subject of abstract or modern algebraic geometry, I recommend the following ICM lectures:
O.Zariski, 1950, vol.2, p.77ff;
B.Segre, 1954, vol.3, p.497ff;
J.P.Serre, 1954, vol.3, p.515ff;
A.Weil, 1954, vol.3, p.550ff;
A.Grothendieck, 1958, p.103.
(This falls obviously under the heading "reading the masters".)
Indeed the whole algebraic geometry session, 1954, vol.3, pp.445-560, has an incredible list of short talks, (Groebner, Hirzebruch, Kodaira, Neron, Rosenlicht, Van der Waerden,...).
the link is: http://www.mathunion.org/ICM/
My apologies for such a brief answer.
The article by Zariski, THE FUNDAMENTAL IDEAS OF ABSTRACT ALGEBRAIC GEOMETRY, points out the advances in commutative algbra motivated by the need to substantiate results in geometry. “The past 25 years have witnessed a remarkable change in the field of algebraic geometry, a change due to the impact of the ideas and methods of modern algebra. What has happened is that this old and venerable sector of pure geometry underwent (and is still undergoing) a process of arithmetization. This new trend has caused consternation in some quarters. It was criticized either as a desertion of geometry or as a subordination of discovery to rigor. I submit that this criticism is unjustified and arises from some
misunderstanding of the object of modern algebraic geometry. This object is not
to banish geometry or geometric intuition, but to equip the geometer with the
sharpest possible tools and effective controls.”
That by Segre argues for the preservation of geometric intuition in algebraic geometry for just this reason, for motivating and suggesting new questions to investigate. It seems particularly articulate and impassioned as he is arguing for a tradition that seems threatened to be lost.
GEOMETRY UPON AN ALGEBRAIC VARIETY
BENIAMINO SEGRE
I. Algebraic geometry — that is to say, the branch of geometry which
deals with the properties of entities represented by algebraic equations — has in
recent years developed in two distinct directions, which in a sense are opposed
to one another. One of these directions is called abstract in as much as it is
concerned with algebraic equations defined over commutative fields subject
only to slight restrictions; here the means employed are purely algebraic,
including in particular ideal theory and valuation theory. The other direction
may properly be called geometrical) this usually deals with algebraic equations
in the complex domain, and from time to time appeals to ideas and methods of
analytic and projective geometry, topology, the theories of analytic functions
and of differential forms.
The dualism between these two disciplines has close relationship and affi-
nity with that which, three centuries ago, arose between l'esprit géométrique of
Descartes and l'esprit de finesse of Pascal, and which, in the past century, on
the one hand divided the geometers into analysts of the school of Plücker and
synthesists of the school of Steiner and, on the other, the algebraists into
purists à la Dedekind and arithmetizers à la Kronecker. However, this dualism,
instead of proving harmful to geometry, offers undoubted advantages when
the two lines of development, with their respective merits and possibilities, are
regarded not as contrasting but as complementary.
We cannot fail to recognise in the abstract method and its technique a
peculiar elegance, an impeccable logical coherence, and to appreciate the im-
portance of the results so far obtained by it, particularly in the study of the
foundations of geometry and the difficult questions concerning the singularities
of algebraic varieties. But equally we cannot fail to recognise that the geometr-
ical approach, with its greater concreteness, lends itself better to the formula-
tion and initial study of new concepts and problems; and that it presents an
incomparable wealth and colour of its own, due to the interweaving of many
diverse strands, to the subtle and perspicuous play of geometrical intuition,
and to the possibility of readily constructing examples and investigating special
cases. We may also point out that, in the geometrical discipline, corresponding
to a more definite notion of algebraic variety, there is a much wider range of
subjects and a far greater number of orientations and contacts with other
important branches of mathematics, which have found, and are finding,
therein inspiration and extensions beyond the purely algebraic field.
Weil’s article describes how arithmetic benefits as well from the algebraization of geometry.
ABSTRACT VERSUS CLASSICAL ALGEBRAIC GEOMETRY
ANDRé WEIL
The word "classical", in mathematics as well as in music, literature or
most other branches of human endeavor, may be taken in a chronological sense;
it then means anything which antedates whatever one chooses to consider as
"modern", and may be used to describe remote antiquity or the achievements
of yesteryear, according to the mood and the age of the speaker. Sometimes, too,
it is purely laudatory and is applied to any piece of work which is thought to be
of permanent value.
Here, however, while discussing algebraic geometry, I wish to use the words
"classical" and "abstract" in a strictly technical sense which will be explained
presently. Until not long ago algebraic geometers did their work exclusively
with reference to the field of complex numbers; at the same time they worked
on non-singular models, or at any rate their concern with multiple points was
merely in order to try to push them out of the way by suitable birational trans-
formations. Thus transcendental and topological tools of various kinds were
available, and it was merely a matter of individual taste, personal inclination or
expediency whether to use them or not on any given occasion. The most deci-
sive progress ever made in the theory of algebraic curves was achieved by
Riemann precisely by introducing such methods. Later authors took consider-
able pains to obtain the same results by other means. In so doing, they were
motivated, at least in part, by the fact that Riemann had given no justification
for Dirichlet's principle and that it took many years to find one. Similarly, the
use of topological methods by Poincaré and Picard, not to mention some more
recent writers, has often been such as to justify doubts about the validity of their
proofs, while conversely it has happened that theorems which had merely been
made plausible by so-called geometrical reasoning were first put beyond doubt
by the transcendental theory.
Now we have progressed beyond that stage. Rigor has ceased to be thought
of as a cumbersome style of formal dress that one has to wear on state occasions
and discards with a sigh of relief as soon as one comes home. We do not ask any
more whether a theorem has been rigorously proved but whether it has been
proved. At the same time we have acquired the techniques whereby our prede-
cessors' ideas and our own can be expanded into proofs as soon as they have
reached the necessary degree of maturity; no matter whether such ideas are
based on topology or analysis, on algebra or geometry, there is little excuse left
for presenting them in incomplete or unfinished form.
What, then, is the true scope of the various methods which we have learnt
to handle in algebraic geometry? The answer is obvious enough. Let us call
"classical" those methods which, by their very nature, depend upon the pro-
perties of the real and of the complex number-fields; such methods may be
derived from topology, calculus, convergent series, partial differential equations
or analytic function-theory. As examples, one may quote the use of the differ-
ential calculus in the proof of the Kronecker-Castelnuovo theorem, of theta-
functions in the theory of elliptic curves and abelian varieties, of topology in the
proof of the "principle of degeneracy". Let us call "abstract" those methods
which, being basically algebraic, are essentially applicable to arbitrary ground-
fields; this includes for instance the theory of differentials of the first, second
and third kinds (but of course not that of their integrals) and the greater part of
the "geometric" proofs of the Italian school. Thus it is plain that, in all cases
where an abstract proof is available, it may be expected to yield more than
any classical proof for the same result. No one could deny this unless he had
made up his mind to ignore fields of non-zero characteristic and was prepared
to maintain that a theorem in algebraic geometry which has been proved for the
field of complex numbers can always be extended to any field of characteristic 0.
There are indeed many cases where this is so; quite often, however, the exten-
sion can only be made to algebraically closed fields. As to denying any existence
to algebraic geometry of non-zero characteristic, not merely would this, in
view of recent developments, amount to denying motion; it would also deprive
algebraic geometry of a rich and promising field of possible applications to
number-theory, where one cannot do without reduction modulo p.
Serre and Grothendieck describe the contribution of cohomology.
I cannot give a good account of this material in a few words, but I strongly advocate reading these articles which marked the introduction of abstract methods in algebraic geometry in its most fruitful period.
