Algebraic geometry 1

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In the 20th century, algebraic geometry experienced three distinct phases. From 1900 to 1930, under the influence of Castelnuovo, Enriques, and Severi, the field expanded significantly, particularly in the study of surfaces, paralleling the earlier advancements in curves. This era established a comprehensive theory of surfaces, exploring connections between synthetic algebro-geometric techniques and topological and analytic methods. However, the abundance of tools and appealing geometric insights sometimes led to shortcuts in proofs and neglect of detailed analysis, particularly in special cases, a common challenge in geometry. The second phase, from 1930 to 1960, saw leaders like Zariski, Weil, and later Grothendieck launch a major initiative to integrate commutative algebra into algebraic geometry. This effort aimed to develop a unified language for discussing projective varieties across different fields, including characteristic p fields and complex numbers. The overarching goal, rooted in Kronecker's vision, was to create a geometry that formally encompassed both arithmetic and projective geometry.