This book is an introduction to algebraic geometry, based on courses given at Rice University and the Institute of Mathematical Sciences of the Chinese University of Hong Kong from 2001 to 2006. The audience for these lectures was quite diverse, ranging from second-year undergraduate students to senior professors in fields like geometric modeling or differential geometry. Thus the algebraic prerequisites are kept to a minimum: a good working knowledge of linear algebra is crucial, along with some familiarity with basic concepts from abstract algebra. A semester of formal training in abstract algebra is more than enough, provided it touches on rings, ideals, and factorization. In practice, motivated students managed to learn the necessary algebra as they went along.

There are two overlapping and intertwining paths to understanding algebraic geometry. The first leads through sheaf theory, cohomology, derived functors and categories, and abstract commutative algebra – and these are just the prerequisites! We will not take this path. Rather, we will focus on specific examples and limit the formalism to what we need for these examples. Indeed, we will emphasize the strand of the formalism most useful for computations: We introduce Gröbner bases early on and develop algorithms for almost every technique we describe. The development of algebraic geometry since the mid 1990s vindicates this approach. The term ‘Groebner’ occurs in 1053 Math Reviews from 1995 to 2004, with most of these occurring in the last five years.