Research Seminar Polytopes and Algebraic Geometry
Christian Haase
Additional information / Pre-requisites
There will be a preliminary meeting March 12 from noon to 2. The room will be announced here.
Comments
In the seminar, bachelor and master students as well as doctoral students will report on their research work for their final theses. The programme will be complemented by guest lectures and the presentation of interesting articles.
The theme this term will be to understand the paper [LLPP]. Possible talks:
Hyperplane arrangements:
- Definitions and examples (coordinate arrangement, braid arrangement, graphical arrangements) [9]
- Counting of regions, intersection lattice, characteristic polynomial. [9]
- Topology of the complement (Betti numbers, cohomology, fundamental group) [8, Chapter 5].
- Wonderful compactifications (Blow up construction, closure construction) [6] or original paper [2]
Matroids:
- Definitions and examples (representable, graphical) [9,5]
- Matroid Fans and their Chow ring, relation to the chow ring of wonderful compactifications [3,5] also original paper [7]
- Different presentations of Chow ring of a fan [3]
K-theory:
- Line bundles, Vector bundles, Chern classes, Whitney formula [4, Chapters 1.4 and 5]
- Chern character, Grothendieck Riemann–Roch [4, Chapter 14]
- K-ring of wonderful varieties [LLPP]
Suggested reading
[LLPP] Larson, Li, Payne, Proudfoot. K-rings of wonderful varieties and matroids, https://arxiv.org/pdf/2210.03169.pdf
[1] Ardila–Mantilla. Intersection theory of matroids: variations on a theme, https://arxiv.org/pdf/2401.07916.pdf
[2] De Concini, Procesi. Wonderful models of subspace arrangements, https://link.springer.com/article/10.1007/BF01589496
[3] Denham. Toric and tropical compactifications of hyperplane complements, https://arxiv.org/pdf/1306.3519.pdf
[4] Eisenbud, Harris. 3264 and All That A Second Course in Algebraic Geometry, https://www.cambridge.org/core/books/3264-and-all-that/
[5] Eur. Essence of independence: Hodge theory of matroids since June Huh, https://arxiv.org/pdf/2211.05724.pdf
[6] Feichtner. De Concini-Procesi wonderful arrangement models - A discrete geometer's point of view, https://arxiv.org/pdf/math/0403183.pdf
[7] Feichtner, Yuzvinsky. Chow rings of toric varieties defined by atomic lattices, https://arxiv.org/pdf/math/0305142.pdf
[8] Orlik, Tarao. Arrangements of Hyperplanes, https://link.springer.com/book/10.1007/978-3-662-02772-1
[9] Stanley. An Introduction to Hyperplane Arrangements, https://www.cis.upenn.edu/~cis6100/sp06stanley.pdf
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14 Class schedule
Additional appointments
Tue, 2024-03-12 12:00 - 14:00Regular appointments
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