Singularities of the general fibers (Zsolt Patakfalvi, EPFL)

14.12.2017 16:15 – 17:15

The fundamental objects of algebraic geometry are algebraic sets, which can be defined as the common zero sets of a few homogeneous polynomials. First, I review how the classification theory of algebraic sets of dimension one differentiates between 3 fundamental cases: parabolic, elliptic, hyperbolic. Then, I explain how in higher dimensions one can only shoot for a decomposition of each algebraic set into parabolic, elliptic or hyperbolic building blocks, instead of having each algebraic set completely in one of the 3 cases. Lastly, I explain how even this fails if the defining polynomials have coefficients in a field of positive characteristic, as in that case general fibers of fibrations between smooth algebraic sets can have singularities. However, this bad behavior seems to be restricted to just a few primes by the global geometry of the parabolic and elliptic building blocks. More generally, a folklore conjecture states that after fixing the dimension, the bad behavior happens only in finitely many characteristics. In the end, I mention how I proved this conjecture with Joe Waldron if the parabolic building block has dimension 2.

During the talk I will assume only at most a knowledge of 2nd-3rd year university courses. In particular, I will explain what algebraic sets are.

PS. The Colloquium will be followed by an aperitif

Lieu

Acacias, Room 17, Colloque

Organisé par

Section de mathématiques

Intervenants

Zsolt Patakfalvi, EPFL

entrée libre

Classement

Catégorie: Colloque