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- W135914604 abstract "In the fifteen years of writing Algebraic Surfaces, Oscar Zariski's work was marked by an extraordinary outpouring of original and creative ideas. Having discovered that many of the classical Italian proofs were incomplete and imprecise, he set to work to develop an abstract theory of algebraic geometry valid over an arbitrary ground field. Abandoning topological and analytical methods, he turned to the new algebra as a means of elucidating basic geometric ideas. Armed with two powerful ideas of commutative algebra, namely, the concept of integral closure and Krull's general valuation rings, he mounted a major attack on the longstanding problem of the resolution of singularities of surfaces and higher dimensional varieties. Many important mathematicians, including Max Noether, Jung, Albanese, and Walker, had worked on the problem, but results had been obtained only in the domain of complex numbers and were partial and incomplete. The general concept of valuation has been introduced to give an ideal-theoretic treatment of the Italian concept of the base conditions imposed by infinitely near points; Zariski applied these results to give an algebraic proof of the theorem that an irrational pencil on an algebraic surface X can have base points only at the singularities of X ." @default.
- W135914604 created "2016-06-24" @default.
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- W135914604 date "1991-01-01" @default.
- W135914604 modified "2023-09-27" @default.
- W135914604 title "THE RESOLUTION OF SOME SINGULARITIES" @default.
- W135914604 doi "https://doi.org/10.1016/b978-0-12-545030-0.50017-6" @default.
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