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Selected Papers, Volume I: On The Classification of Varieties and Moduli Spaces

David Mumford
Springer Verlag
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Darren Glass
, on

There is little doubt that David Mumford is one of the most influential mathematicians of the second half of the twentieth century. Among Mumford's many contributions were the first purely algebraic proof of the existence of a moduli space of curves of a fixed genus and the extension of the Enriques-Kodaira classification of complex surfaces to algebraic surfaces in characteristic p.

Springer has recently released Selected Papers: On The Classification of Varieties and Moduli Spaces, a collection of 30 of the 51 papers that Mumford wrote in algebraic geometry (this is in addition to 41 papers from the research he has been doing on vision for the last two decades, 13 books, and a handful of other items). The papers are divided into three sections, each of which comes with commentary and annotation by Mumford and other mathematicians, which give nice summaries and introductions to much of his work.

But the bulk of the 800 pages in the collection is dedicated to the papers themselves. The first section of the book collects a number of papers on his work on geometric invariant theory, a subject which was a major area of research in the 19th century, and which Mumford used in order to construct moduli spaces and prove many results on the enumerative combinatorics of these spaces. The second section moves from the moduli space of curves to that of abelian varieties, and in particular the papers written by Mumford on the relationship between Theta functions and abelian varieties. The third and final section contains Mumford's papers on the classifications of surfaces and higher dimensional varieties.

Much more could (and has) been written about the influence of Mumford's work. Reading these papers is exciting both for their mathematical content and to watch the evolution of the ideas they contain over the decades. The level of mathematical exposition is also high, although given the depth and breadth of the results contained within, that is only a secondary reason to have this book. Now that it is on my shelf, I imagine I will save many trips to the library to look up the papers contained inside, and I imagine that it is a book that most algebraic geometers — and all libraries — will not want to do without.

 Darren Glass is Assistant Professor of Mathematics at Gettysburg College.

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