## Seminarium "Fundamenty Geometrii algebraicznej" 2021/22

### Maria Donten-Bury, Joachim Jelisiejew

I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado. [A. Grothendieck]

Usoswebowa strona

Podręcznik "The Rising Sea" Raviego Vakila

Opis seminarium: referaty na podstawie powyższej książki; nie zakładamy żadnej wiedzy z geometrii algebraicznej (i początkowo niewiele wiedzy z algebry; w szczególności można wziąć to seminarium równolegle z algebrą przemienną. W razie pytań proszę pisać.

### Opis spotkań

Pierwsze spotkania w październiku 2021.
1. 4.10 JJ: general big picture and intro. Sheaves.
2. 11.10 PO: presheaves and sheaves, pushforwards, sheaf from sheaf on a base, inverse image sheaf (Chap 2).
3. 18.10 PO (continued). RS: Spec of a ring, distinguished opens (Chap 3), structure sheaf of affine scheme (Chap 4.1), definition of the scheme in general.
4. 25.10 KJ: in particular affine-local properties, affine communication lemma.
5. 8.11 MJ: morphisms of schemes I: definition, morphisms to affine scheme, examples.
6. 15.11 JZ: projective schemes I.
7. 22.11 JZ: projective schemes II.
8. 29.11 WD: tensor products.
9. 6.12 KJ: normalization.
10. 13.12 PO: images.
11. 20.12 RS: separated morphisms.
12. 10.01 RS: separated morphisms -- second part.
13. 17.01 MJ: proper schemes and morphisms.
14. 24.01 ML: group schemes and rigidity lemmas.
15. 28.02 WD: dimension of fibres.
16. 7.03 JZ: smoothness and regularity I (around Chapter 12).
17. 14.03 RS(?): smoothness and regularity II.
18. 21.03 PO: Line bundles.
19. 28.03 MJ: Quasicoherent sheaves on projective schemes.
20. 4.04 MJ: continued. WD: Pushforwards and pullbacks of quasicoherent sheaves.
21. 11.04 WD: Pushforwards and pullbacks of quasicoherent sheaves. ML: morphisms of P^n
22. 25.04 RS: Ample line bundles and morphisms to projective spaces
23. 9.05 JZ: Relative versions of Spec and Proj, projective morphisms
24. 16.05 KJ: Basic properties of Cech cohomology
25. 23.05 KJ (continued), MJ: Crucial tools for curves
26. 31.05 MJ (continued), ML: Curves of small genus and hyperelliptic curves
27. 6.06 ML (continued), PO: Genus 1 curves
28. 13.06 PO (continued)