The Resolution of Some Singularities

In the fifteen years that followed the writing of Algebraic Surfaces, 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 wholeheartedly to the new algebra as a means of elucidating basic geometric ideas.

Armed with two powerful ideas of commutative algebra—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.