Math 663: Algebraic Geometry 1.
Algebraic Geometry 1.
Math 676 or concurrent enrollment.
Basic definitions and theorems on varieties, sheaves, and schemes.
Desired Learning Outcomes
Math 676 or current enrollment
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.
- Serre varieties
- Sheaves and affine varieties
- Definition of varieties and morphisms
- Products and separation
- Fibres of a morphism
- Complete varieties
- Relation with complex analytic varities
- Affine schemes
- Relationship between varieties and schemes
- Closed subschemes
- Functor of points
- Proper varieties and finite morphisms
Possible textbooks for this course include (but are not limited to):
- D. Mumford, The Red Book of Varieties and Schemes, Second, Expanded Edition
If the above outline is followed, there will probably not be room for additional topics. One could possible expand on applications of schemes to arithmetic.
Courses for which this course is prerequisite