This will be an introductory course. The formal prerequisite is a first year graduate course in algebra, such as Math 8200/1/2. Other knowledge which can help one to understand important concepts in algebraic geometry includes (i) commutative ring theory and (ii) various facts from the Manifolds and Topology course ( Math 8300/1/2). During the Fall and Winter, however, the course will be taught in a way that will keep the amount of required previous knowledge in both of those subjects to a minimum. In particular, some material about commutative rings will be presented in the lectures when it is needed for the main subject matter.
The textbook during the Fall and Winter Quarters will be Algebraic Geometry: A First Course, by Joe Harris. While it is clearly a graduate text, the author has succeeded in minimizing the technical algebraic prerequisities. To a close approximation, we will study topics from Part I of the text (Examples of Varieties and Maps) during the Fall Quarter, and topics from Part II of the text (Attributes of Varieties) during the Winter Quarter. This means that we will study some substantial examples before developing extensive amounts of theory. This approach may be somewhat unusual in 20-th century mathematics, but it works well for learning algebraic geometry - especially if one wants to have a good intuitive grasp of what the theorems really mean.
During the Spring quarter, the subject matter will be an introduction to the theory of schemes, so that there will be a different textbook. If we achieve our goals for the first two quarters, then it will be reasonably clear why we should study schemes in order to answer certain natural questions about varieties.
The text will be supplemented by material from various other sources. Thus, a list of references will be provided, and copies of lecture notes from previous years will be available.
The required work in the course will consist of solving exercises from the text and writing up the solutions. One office hour each week will be structured as an informal seminar, where students can not only ask questions but can also participate in finding the answers.
Last updated Wednesday, April 8, 1998.
Joel Roberts
351 Vincent Hall
625-1076
e-mail:
roberts@math.umn.edu
http://www.math.umn.edu/~roberts
