Math 561: Intro to Algebraic Geometry 1
Introduction to Algebraic Geometry 1.
Math 571 or concurrent enrollment.
Basic definitions and theorems on affine, projective, and quasi-projective varieties.
Desired Learning Outcomes
Math 671 or concurrent 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.
- Algebraic plane curves
- Rational curves
- Relation with field theory
- Rational maps
- Singular and nonsingular points
- Projective spaces
- Affine varieties
- Affine space and the Zariski topology
- Regular functions
- Regular maps
- Rational functions and rational maps
- Quasiprojective varieties
- The Zariski topology on projective space
- Regular and rational functions
- Products and maps of quasi-projective space
- Definition of products
- Properness of projective maps
- Finite maps
- Definition of dimension
- Dimension of intersection with a hypersurface
- Dimension of fibres
- Application to lines on surfaces (optional)
Possible textbooks for this course include (but are not limited to):
- I. R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space
If time permits, additional topics may be covered. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves.