Algebraic Geometry often seems very abstract, but in fact it is full of concrete examples and problems. This side of the subject can be approached through the equations of a variety, and the syzygies of these equations are a necessary part of the study. This book is the first textbook-level account of basic examples and techniques in this area. It illustrates the use of syzygies in many concrete geometric considerations, from interpolation to the study of canonical curves. The text has served as a basis for graduate courses by the author at Berkeley, Brandeis, and in Paris. It is also suitable for self-study by a reader who knows a little commutative algebra and algebraic geometry already. As an aid to the reader, the appendices provide summaries of local cohomology and commutative algebra, tying together examples and major results from a wide range of topics.

Written for: Graduate students, research mathematicians

• Preface: Algebra and Geometry
• Free Resolutions and Hilbert Functions
• First Examples of Free Resolutions
• Points in P²
• 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
• Appendix 1: Introduction to Local Cohomology
• Appendix 2: A Jog Through Commutative Algebra