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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 dis- covery 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.

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.

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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.