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 and Chapters 1-4 and 8 of Eisenbud’s Commutative Algebra.

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: 10:30-11:30am (Waterloo time) MWF, MC 5479

Office hours: W 11:30-12:30, F 2-3 ****(in-person), by appointment (virtual)

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).

HW:

HW 1, due Jan 17 crowdmark discord solutions

HW 2, due Jan 26 crowdmark discord solutions

HW 3, due Feb 7 crowdmark discord

HW 4, due Feb 14 crowdmark

Final presentation: There’s no need to think about the final presentation before break, but there is a topics list if you want to peruse it.

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 presheaves (Vakil 2.1)

Jan. 8: Sheaves (Vakil 2.2)

Jan. 10: Abelian categories; sheaves as a category and sheafification (Vakil 1.5, 2.3-7)

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