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

entrée libre