Geometry of Syzygies

The author taught at Brandeis University for twenty-seven years, with sabbatical time spent in Paris, Bonn, and Berkeley, and became Director of the Mathematical Sciences Research Institute in Berkeley in the Summer of 1997. At the same time he joined the faculty of UC Berkeley as Professor of Mathematics. In 2003 he became President of the American Mathematical Society. He currently serves on several editorial boards (Annals of Mathematics, Bulletin du Société Mathématique de France, Springer-Verlag's book series Algorithms and Computation in Mathematics).

From the reviews: "This book is devoted to offer ... an approach to the study of this algebraic subject (syzygy = relation among generators of a module) ... . a student would learn a lot of algebraic geometry from it. The double bet of the book is to be able to be a complete textbook ... and at the same time to become a useful reference text for research work on the subject. I would say that both aspects of the bet have been gained ... ." (Alessandro Gimigliano, Zentralblatt MATH, Vol. 1066, 2005) "This book may be regarded as a complement to the author's Commutative Algebra ... . It begins by explaining syzygies and their connection with the Hilbert function, and turns to describing various aspects of algebraic geometry ... . Two appendices provide the background in commutative algebra and local cohomology. Together with exercises, it gives a good survey of topics often not covered." (Mathematika, Vol. 52, 2005) "This monograph is devoted to the geometric properties of a projective variety corresponding to the properties of its syzygies ... . Altogether, this is a most welcome addition to the literature and will help many a reader bridge the gap between the abstractions of algebra and the more tangible field of geometry." (Ch. Baxa, Monatshefte fur Mathematik, Vol. 150 (1), 2006) Aus den Rezensionen: "... Das vorliegende Buch beschaftigt sich mit der qualitativen geometrischen Theorie der Syzygien. ... Es gibt zwei sehr kompakt geschriebene Anhange: Der erste fuhrt in die lokale Kohomologie ein, der zweite stellt fur das Buch notige Vorkenntnisse der kommutativen Algebra ... zusammen. Dieses Buch ist sehr elegant geschrieben und vermittelt viele interessante Ideen. In der Lehre konnte es gut fur weiterfuhrende Vorlesungen uber algebraische Geometrie verwendet werden." (Franz Pauer, in: Internationale Mathematische Nachrichten, December/2009, Issue 12, S. 45) "This very interesting book is the first textbook-level account of syzygies as they are used in algebraic geometry. ... The reader will find two very good and useful appendices. ... The book can be read, without any problem, by a student who has received already a little introduction in commutative algebra and algebraic geometry. I highly recommend this nice and deep textbook for all students and researchers studying algebraic geometry or commutative algebra." (Dominique Lambert, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)

• Free Resolutions and Hilbert Functions.- First Examples of Free Resolutions.- Points in ?2.- Castelnuovo-Mumford Regularity.- The Regularity of Projective Curves.- Linear Series and 1-Generic Matrices.- Linear Complexes and the Linear Syzygy Theorem.- Curves of High Degree.- Clifford Index and Canonical Embedding.