Seminar - Concrete Algebraic Geometry SS 2023

Teacher: Daniele Agostini.

Time: to be determined.

Grade: you will be asked to give a presentation and prepare a short handout. The grade will be based on both, but the presentation has more relevance.

Announcements

• There will be a first meeting on Wednesday, February 1 at 14:15 in my office.
• Please register to the seminar on URM.

Content

The aim of this seminar is to present some concrete applications in algebraic geometry, through some of its many beautiful examples and theorems. A preliminary list of topics is below, but they can also be changed according to the interests of the students. Each topic would ideally be prepared by two students and presented in two separate lectures.

Preliminary list of topics

• The 27 lines on a smooth cubic surface: Chapter 5 of [Hulek]. This can be presented in two lectures and some background in algebraic geometry would be desirable.
• Biduality and Plücker formulae for plane curves: Section 2 of Chapter 1 of [GKZ]. This can be presented in two lectures and requires less background in algebraic geometry.
• Equations and syzygies of the rational normal curve: Chapter 1 and Sections 6A and (possibly) 6B of [Eis]. This can be presented in two lectures and is ideal for people that like commutative algebra.
• Water waves and elliptic curves: Section 1 of Chapter 5 of [Kir] plus some facts in Chapter 4 of [Kas]. This can be presented in two lectures, and requires only background in complex analysis. Ideal for physically inclined people.
• Algebraic geometry and statistics: the First Lecture in [HS] and something of the second. Can be presented in two lectures. This does not require a lot of background in algebraic geometry, if we take some things or granted.
• Harnack's Theorem: Section 11.6 of [BCR]. One can also look at Harnack's original paper [Har]. This can be presented in one lecture and does not require a lot of algebraic geometry. Good if you like the real numbers.
• ...

References : The references indicated above are:

• [BCR] J. Bochnak, M. Coste, M-F. Roy, Real algebraic geometry.
• [Eis] D. Eisenbud, The geometry of syzygies.
• [GKZ] I. Gelfand, M. Kapranov, A. Zelevinsky, Discriminant, Resultants and Multidimensional Determinants.
• [Hul] K. Hulek, Elementare Algebraische Geometrie. There is also a version in English.
• [HS] J. Huh, B. Sturmfels, Likelihood Geometry.
• [Kir] F. Kirwan, Complex algebraic curves.
• [Kas] A. Kasman, Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs.

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