Topics in Algebraic Geometry, Spring 2016

Teacher: Bas Edixhoven.

Time: Wednesdays, 11:00--11:45 (talk 1), 12:00--12:45 (talk 2), 13:00--13:40 (discussion/problem session).

Place: room 401, Snellius.

All students writing their master theses in the Spring of 2016 in the area of algebra, geometry and number theory, or planning to start doing so in the near future, are advised to participate.

About half of the course will take place in the form of a seminar, where those who wish to obtain credit points will give lectures.

The rules for obtaining 6 EC for this seminar are as follows:

• speak at least two times;
• not too much of the same;
• there must be some originality in the talk (e.g., at least work out an example of yourself, or state and prove your own new theorem!);
• students provide notes for their own lectures.

Up to now, the following students have expressed their interest in this course:

• Arthur Bik (UL); stopped
• Franco Giovenzana (UL, 6 EC needed)
• Luca Giovenzana (UL, 6 EC needed)
• Pol van Hoften (UU)
• Andrea Marrama (UL) (3 EC needed)
• Nikitas Nikandros (UU)
• Job Rood (UL)
• Marius Stekelenburg (UL)
• Raoul Wols (UL, 3 EC needed).

Given the interest of these participants, the following topics will be on the program:

• Etale cohomology
• Abelian varieties (over C, over finite fields, complex multiplications)
• Jacobian varieties (construction of, for etale cohomology of curves)
• something related to the master thesis subjects for Luca and Franco Giovenzana and Andrea Marrama?

February 4. First meeting (this was on Thursday; we decided to move to Wednesday).

• 9:00: Organisational part.
• 10:00: Bas Edixhoven: Algebraic definition of some singular cohomology groups (see also page 194+ of this text in Dutch).

February 10.

• 11:15: Bas Edixhoven: Smooth and etale morphisms; here the notes,
• 12:15: Bas Edixhoven: Smooth and etale morphisms.

February 17.

• 11:15: Pol van Hoften: Sites, sheaves, with special emphasis on the etale site of a scheme; notes,
• 12:15: Raoul Wols: The etale fundamental group; notes.

February 24. We had a guest: Rico Apon. Arthur Bik stopped.

• 11:00: Job Rood: The lattice of a complex elliptic curve cannot be algebraically defined; notes,
• 12:00: Bas Edixhoven: Some examples of sheaves on the etale site of a scheme, Kummer sequence; notes.

March 2. Job will not be there. Nikitas could not reach Leiden because of train problems. We did exercises from 11 until 12, and a bit of discussion after 13:00.

• 12:00: Marius Stekelenburg: Sheafification and injective resolutions; notes,

March 16.

• 11:00: Nikitas Nikandros: Start of jacobian varieties of curves; notes,
• 11:00: Luca Giovenzana: Torsors and degree one cohomology; notes.

March 30. Luca and Franco are in Orsay.

• 11:00: Pol van Hoften: fpqc descent of morphisms; notes,
• 12:00: Andrea Marrama: quotients of schemes by finite group actions, symmetric powers of curves; notes.

April 13.

• 11:00: Bas Edixhoven: more on quotients of schemes by finite groups; notes (for both parts),
• 12:00: Bas Edixhoven: more on the representability of the relative Picard functor for a family of curves.

April 20.

• 11:00: Bas Edixhoven: descent of quasi-coherent modules, comparison of H^1 of G_m for several topologies; see pages 4--6 of notes,
• 12:00: Raoul Wols: the n-torsion of abelian varieties over algebraically closed fields; notes.

May 4.

• 11:00: Job Rood: H^2 of curves over algebraically closed fields and the Brauer group; notes,
• 12:00: Franco Giovenzana: spectral sequences, Grothendieck spectral sequence; notes.

May 11.

• 11:00: Luca Giovenzana: examples of spectral sequences, Leray, Hochschild-Serre; notes,
• 12:00: Marius Stekelenburg: constructible sheaves and towards the proper base change theorem, following SGA 4.5, pages 39--51; notes.

May 18.

• 11:00: Nikitas Nikandros: towards the proper base change theorem, following Stacks Project 0A5F,
• 12:00: Bas Edixhoven: proper base change theorem, following Stacks Project 095S; notes,
• 13:00: Franco Giovenzana: algebraic interpretation of the functional equation of Riemann's theta function.

May 25. Note that we start at 14:00, and we are then in room 408.

• 14:00: Bas Edixhoven: end of the discussion of the proper base change theorem; see pages 2-3 of notes,
• 15:00: Bas Edixhoven: compactly supported cohomology; notes,
• 16:00: Bas Edixhoven: sketch of l-adic cohomology, Weil conjectures, Lefschetz trace formula; notes.