Description: This is a second course on algebraic geometry, covering basics of the theory of schemes and sheaves, including affine and projective schemes, morphisms between schemes, fibered products, dimension and regularity, line bundles, Riemann-Roch and Serre duality for curves.

Course outline:

Prerequisites: A basic knowledge of commutative algebra and varieties in $\mathbb{C}^n$. One of PMATH 446 or 464 should be sufficient, though both would be better. I’ve been recommending to students who want to be sure they have the background to go through Chapters 1,2,4 and 6 of Fulton’s Algebraic Curves

Course format: This course will be offered through Fields Academy, and thus can be taken in person on campus or online. More details about taking the class online will be available closer to the starting date.

Instructor: Ben Webster ([email protected])

Lecture times:

Text: The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil; I will also probably make some reference to *The Geometry of Schemes* by Eisenbud and Harris (Waterloo students can download for free from Springer).

Marking: There will be regular problem sets, which will be turned in and marked on Crowdmark. For a final project, students will record a presentation of an additional topic of their choosing.

Schedule: I’ll add a schedule below once I know course times and have had a chance to plan.

Jan. 6: General motivation; definition of sheaves (Vakil 2.1-2)

Jan. 8: Abelian categories; sheaves as a category and sheafification (Vakil 1.5, 2.3-4)

Jan. 10: Gluing sheaves and inverse image (Vakil 2.5-7)

Jan. 13: Affine schemes I (Vakil 3.1-3)

Jan. 15: The Zariski topology (Vakil 3.4-7)

Jan. 17: Affine schemes II (Vakil 4.1-2)

Jan. 20: Not-so-affine schemes (Vakil 4.3-4)

Jan. 22: Proj and projective schemes (Vakil 4.5)

Jan. 24: Properties of schemes I (Vakil 5.1-4)

Jan. 27: Quasicoherent sheaves (Vakil 6.1-4)

Jan. 29: Support and associated primes (Vakil 6.6)