3 credits
Fall 2026 Lecture Upper Division
This is the first course in a two-semester introductory sequence in algebraic geometry. This first course focuses on the study of algebraic varieties over algebraically closed fields. Topics include affine and projective varieties, morphisms and rational maps, nonsingular varieties, lines on and intersection in projective space. Additional topics vary by semester and may include Luroth's theorem, elimination theory, Grassmannians, the 27 lines on a smooth cubic surface, and a first look at sheaves and schemes. Prerequisites: MA 55300, MA 55400, MA 57100, and MA 55700 (can be taken concurrently). MA 56200 and MA 57200 are helpful but not required.
Learning Outcomes1Define and prove basic properties of affine and projective varieties.
2Compute explicit examples of algebraic varieties and morphisms/rational maps between them.
3Study singularities and their classification, for example for plane curves.
4Use Bezout's theorem to compute intersection numbers.
5Apply the theory developed in the class to prove classical theorems in algebraic geometry, for example Luroth's theorem, Pascal's theorem, and the Cayley-Salmon theorem stating there are exactly 27 lines on a smooth cubic surface.
6Investigate additional topics as time permits.
Restrictions
made by Purdue students.