Toric geometry – ETHZ – Spring 2025
Lecturer
Lectures. Thursdays 16:15–18:00, HG E 41. The first lecture is on Thursday, February 20, 2025.
Office hours. Wednesday, 15:30–18:00 or by appointment (office HG J 16.2).
Course description. Toric varieties provide a rich class of examples in algebraic geometry that bridge combinatorics and geometry, making them an ideal starting point for exploring the interplay between these fields. We will introduce toric varieties, study their combinatorial and abstract structures, and examine their basic geometry, including singularities, Picard groups, and cohomology.
Prerequisites. Some background in algebraic geometry is highly desirable. In the absence of an algebraic geometry background, knowledge of at least one of differential, complex, or symplectic geometry is required.
References.
J.-P. Brasselet. Introduction to Toric Varieties, Course notes, 2008
W. Fulton. Introduction to Toric Varieties. Princeton University Press, 1993
D. A. Cox, J. B. Little, H. K. Schenck. Toric Varieties. American Mathematical Society, 2011
Reading for specialized topics will be assigned as we progress.
Course log.
A brief description of each lecture's content will appear here.
20 Feb 2025. Organization of the seminar. Introduction and motivation.
27 Feb 2025. Cones, faces, monoids. Algebraic varieties.
6 Mar 2025. Affine toric varieties [interactive example: double cone].
13 Mar 2025. From fans to toric varieties.
20 Mar 2025. Orbits and their closure.
27 Mar 2025. Geometric properties [example of complete, non-polytopal fan, by Sirawit]. From toric varieties to fans.
3 Apr 2025. Polytopes. Divisors.
10 Apr 2025. Class and Picard groups. Introduction to sheaf cohomology.
17 Apr 2025. Class and Picard groups revisited. Line bundles on toric varieties.
8 May 2025. Singularities and their resolutions.
15 May 2025. Chow groups, characteristic classes, Riemann–Roch and Pick's formula.
22 May 2025. Stanley's theorem.
The g-conjecture, a generalisation of Stanley's theorem. It has been proved by Adiprasito in Dec. 2018.
Riemann surfaces – ETHZ – Spring 2024
Lecturer
Lectures. Thursdays 16:15–18:00, HG D 5.2. The first lecture is on Thursday, February 22, 2024.
Office hours. By appointment (office HG J 16.2).
Course description. The course will be a first introduction to Riemann surfaces. These are beautiful objects that sit at the intersection of algebra, geometry, and analysis. We will aim to cover the theorems of Riemann–Hurwitz and Riemann–Roch, as well as the basics of Hurwitz theory. Time permitting, we may delve into additional subjects such as abelian integrals and the Abel–Jacobi theorem.
Prerequisites. Theory of functions of one complex variable, basics of topology. Familiarity with the theory of smooth manifolds and algebraic topology would be useful, but not necessary.
References.
E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris. Geometry of Algebraic Curves, Volume 1. Springer–Verlag, 1985
R. Cavalieri, E. Miles. Riemann Surfaces and Algebraic Curves. Cambridge University Press, 2016
O. Forster. Lectures on Riemann Surfaces. Springer–Verlag, 1981
Course log.
A brief description of each lecture's content, together with some notes, will appear here.
22 Feb 2024. Presentation of the course. Real smooth vs. analytic. Holomorphic functions. Exercise sheet 1, Solutions
29 Feb 2024. The problem of multi-valued functions. Manifolds. Definition of Riemann surfaces. Exercise sheet 2, Solutions
7 Mar 2024. Maps between manifolds. Classification of topological compact surfaces. Exercise sheet 3, Solutions
14 Mar 2024. Classification of complex tori, plane affine and projective curves. Exercise sheet 4, Solutions
21 Mar 2024. Elliptic curves. Multiplicity of holomorphic maps at a point.
28 Mar 2024. Riemann–Hurwitz formula, genus–degree formula, meromorphic functions. Exercise sheet 5, Solutions
11 Apr 2024. Meromorphic functions on the Riemann sphere and the tori. Exercise sheet 6, Solutions
18 Apr 2024. Divisors, principal divisors, Picard group, linear spaces of meromorphic functions. Exercise sheet 7, Solutions
25 Apr 2024. Finiteness theorem. Meromorphic forms, canonical divisors, Serre duality. Exercise sheet 8, Solutions
2 May 2024. Residue theorem. Riemann–Roch theorem. Exercise sheet 9 (solutions can be found in the notes)
16 May, 2024. Abel–Jacobi theory.
23 May, 2024. Abel–Jacobi theory (continued), Hurwitz numbers, monodromy representation.
30 May, 2024. Hurwitz numbers and permutations. Exercise sheet 10, Solutions
Exam. The exam is a 20 minute oral exam. he next exam session is scheduled for January 27, 2025.
The first question on your exam will be chosen randomly from this collection of questions.
A nice video hinting at some of the points explained in the lectures.
Linear Algebra (in Italian) – UTrieste – Fall 2016
Exercise classes