09 - London

The 9th meeting was in King's College on 24 March 2014.

The schedule will be:

12:00 - 1:00 Lunch

 1:00 - 2:00 
Strand building, Room S0.13
                    Jérémy Blanc (Universität Basel)

2:15 - 3:15 Strand building, Room S-1.04
Nick Shepherd-Barron (King's College, London)

3:15 - 3:30 Coffee break

3:30 - 4:30 Strand building, Room S-1.04
Sergey Galkin (HSE, Moscow)

Titles and Abstracts

Jérémy Blanc

Title: Dynamical degrees of birational transformations of projective surfaces

The dynamical degree λ(f) of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n-th iterate of f. I will describe the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of λ(f) and the structure of the conjugacy class of f. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed, well ordered set of algebraic numbers.

Sergey Galkin

Title: Cubics: lines, squares, and irrationality
Abstract: I will describe our joint work with Evgeny Shinder. We prove that generic cubic fourfold is irrational under the assumption that the class of an affine line is not a zero divisor in the Grothendieck ring of complex varieties. Main new geometric ingredient of the proof is a beautiful formula, that relates classes of a cubic hypersurface itself, its symmetric square, variety of lines and the singular locus. The formula is unconditional and holds over any reduced cubic hypersurface over arbitrary field. It also gives another proof for the theorem of Cayley about 27 lines on a surface, as well as many similar results for singular surfaces over non-algebraically closed fields.​

Nick Shepherd-Barron

Title: Some effectivity questions in the Cremona group.

Abstract: Some special Cremona transformations appear bot as examples of the most accessible not not-trivial invertible dynamical systems (Henon maps) and as the essential building blocks of certain encryption algorithms (Feistel ciphers). This motivates the main point of my talk, which is a discussion of how quickly we can estimate quantities such as the entropy of a transformation that is givn by explicit polynomial formulae.