Instructional Workshop
Surfaces: Geometry and Arithmetic

Monday 14 April--Friday 18 April

at the

Mathematics Research Centre, University of Warwick, UK

Organizers/Instructors

Martin Bright, Ronald van Luijk, Samir Siksek, Damiano Testa

Format

There will be three instructional lectures every morning. In the afternoon the participants will have the opportunity to work on exercises in groups.

Programme pdf

Monday:

Bright: Picard groups, definition and simple examples (P^2, quadric surface in P^3) notes (pdf)
van Luijk: Canonical divisor on hypersurface of degree d in P^n and on complete intersections; very ample divisors notes (pdf)
Testa: Classification and the minimal model program; the Hodge diamond notes (pdf)
Exercises 1 (pdf)

Tuesday:

Bright: Riemann-Roch on surfaces, genus and effectivity of (-1)-curves, adjunction formula
van Luijk: Picard groups of del Pezzo surfaces notes (pdf)
Testa: Root Lattices and their automorphism groups; Segre-Manin for del Pezzo surfaces I notes (pdf)
Exercises 2 (pdf)

Wednesday:

Testa: Segre-Manin for del Pezzo surfaces II
van Luijk: Growth of rational points
Bright: Explicitly computed examples of Brauer-Manin obstruction through quadratic reciprocity notes (pdf)

Thursday:

Testa: Central simple algebras and Brauer groups notes (pdf)
Bright: Reformulate easy examples the enlightening way notes (pdf)
van Luijk: Galois cohomology I
Exercises 3 (pdf)

Friday:

van Luijk Galois Cohomology II
Bright and Testa: How to find the algebra from Galois cohomology? Magma Program

Prerequisites. Some familiarity with basic algebraic and arithmetic geometry is assumed, at the level of the first two chapters of Silverman's book "Arithmetic of Elliptic Curves".

Preliminary Reading. Participants may find it helpful to work through the first few chapters of "Chapters on Algebraic Surfaces" by Miles Reid (available online). Preliminary reading on group cohomology will also be useful, a good reference is chapter 2 of Milne's notes on Class Field Theory.

Registration is now closed

Particpants from universities belonging to the GTEM network might be able to obtain travel and accommodation expenses from GTEM; please contact your local scientist-in-charge.

This event is part of the April–June Active Period in Arithmetic Geometry and the 2007–2008 Warwick EPSRC Symposium on Algebraic Geometry

Samir Siksek

Instructional Workshop Surfaces: Geometry and Arithmetic