Lectures: 2 sessions / week, 1.5 hours / session
Mumford, David. The Red Book of Varieties and Schemes. Vol. 1358, Lecture Notes in Mathematics. New York: Springer-Verlag. ISBN: 354063293X.
Includes the Michigan Lectures (1974) on Curves and Their Jacobians.
There will be weekly homework.
The course grades will be based on weekly homework and a take home final as shown in table.
Students should have some familiarity with commutative algebra and basic topology. Some more advanced algebraic topology may also be useful as might some knowledge of category theory.
The aim of this course is to introduce students to some basic notions and ideas in algebraic geometry, paving the way for a study of Grothendiecks's theory of schemes (second semester). Though the theory of schemes and cohomology is generally accepted as the "right" setting for algebraic geometry, these subjects require a substantial amount of technical language which to the uninitiated can sometimes obscure the beauty and elegance of the subject. Thus in an effort to make the subject more accessible and to give students basic techniques without delving into a morass of technical details, I will not use this language in this course but rather the more classical language of varieties, which, though perhaps lacking the elegance of schemes, is perfectly adequate for many situations in algebraic geometry. Thus the goal of the course is more to give students a feeling for algebraic geometry, rather than to develop the foundations of the subject, which students should learn in subsequent courses on schemes.
My plan is to follow somewhat loosely the first chapter of Mumford's book at the begining of the course, and then to discuss the appendix on curves and their Jacobians. Other topics may be included as time permits.
Examples with emphasis on algebraic curves and surfaces are developed. The course may be taken concurrently with 18.705, Commutative Algebra. Knowledge of elementary algebraic topology and elementary differential geometry is recommended, but not required.