Topic: This course is an introduction to the theory of algebraic curves and surfaces. An important aim of the course is to develop geometric intuition while using the language of schemes developed in the basic algebraic geometry course, thus building a solid foundation for further study.
Lectures: All the lectures will take place in MA A 3 30, 10.00-12.00 on Mondays. I will upload handwritten notes for each lecture.
Exercise Sessions: There is a weekly exercise session in CE 1 100 running on Mondays, 14.00-16.00, led by Kamil Rychlewicz. During the exercise session, you can work on the exercise sheets and related problems or ask any questions related to the course.
Exercise sheets: A number of exercises will be assigned mainly with the goal of keeping you up to speed with the material. I will assign roughly 5 problems each week. You should attempt all the exercises as the final oral exam will likely resemble these questions. Asking me, Kamil, and each other questions about the problems is highly encouraged.
If you would like feedback, you can submit your solutions to the exercises up to two weeks after they are assigned.
You are also welcome to work together on the solutions of the exercises. However, please do not submit multiple copies of the same solution for feedback.
Office hours: Tuesdays 16:30-18:00 in MA B3 494. If no one is around at 17:15 I might leave.
Exam: There will be an oral exam accounting for 100% of the grade. The location and precise times will be announced later.
References:
We will mainly follow Chapters IV and V in
- R. Hartshorne, Algebraic Geometry.
Other good references are:
R. Vakil, Fundamentals of Algebraic Geometry.U. Görtz and T. Wedhorn, Algebraic Geometry I & II.
Definition of varieties. Recollection of relevant material from the language of schemes: integral, reduced, separated, proper, finite type, projective.
Weil divisors, class groups, and line bundles.
References: [Hartshorne, II.6 and IV.4] [Vakil, 15.4]
Sheaf cohomology recap. Statement of Serre duality.
References: [Vakil, 18] [Hartshorne, III.2-5]
Line bundles on curves. Riemann--Roch
References: [Hartshorne IV.1] [Vakil, 19.1,19.2]
Maps between curves and Riemann--Hurwitz. Hyperelliptic curves.
Reference: [Hartshorne, IV.2]
Curves of genus one.
Reference: [Vakil, 19.9] [Hartshorne,IV.4]
Embedding curves in projective spaces. Projective geometry.
Reference: [Hartshorne, IV.3] [Vakil, 18.6, 13.4]
Cohomology and base change
Reference: [Vakil, §24 & §25]
Intersection theory on surfaces
Reference: [Hartshorne, V.1] [Beauville, I]
Ruled surfaces
Reference: [Hartshorne, V.2]
Blow ups of points on surfaces
Reference: [Hartshorne, V.3] [Vakil, 22]
Birational maps between surfaces
Reference: [Hartshorne, V.3 & V.5]
Contracting curves. Castenuovo's Criterion. Minimal surfaces.
[Hartshorne, V.5]
Cubic surfaces
Reference: [Hartshorne, V.4] [Beauville, IV] [Vakil, 27]