Students should achieve mastery of the topics listed in the minimal learning outcomes on the Math 561 Wiki page. 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.

1. Algebraic plane curves
2. Rational curves
3. Relation with field theory
4. Rational maps
5. Singular and nonsingular points
6. Projective spaces
7. Affine varieties
8. Affine space and the Zariski topology
9. Regular functions
10. Regular maps
11. The Zariski topology on projective space
12. Products and maps of quasi-projective space
13. Properness of projective maps
14. Dimension.