Séminaire de géométrie algébrique

Virt: Tomasz Pełka (Bilbao): Smooth affine factorial surfaces of Kodaira dimension zero with trivial units

Europe/Paris
001 (batiment I)

001

batiment I

Département de mathématiques Bâtiment I Faculté des Sciences 2 Boulevard Lavoisier F-49045 Angers cedex 01 France
Description

We consider complex affine surfaces S_{p,q} given in C^4 by
{yx^d=z-p(x), wz^e=x-q(z)}, where p,q are polynomials of degrees d,e;
p(0)=q(0)=1. Using these surfaces as a simple example, we introduce
various notions in algebraic geometry and topology. First, we compute
their standard boundaries, showing that S_{p,q} is isomorphic to
S_{p',q'} if and only if {p,q}={p',q'}. Next, applying calculus of graph
manifolds to tubular neighborhoods of these boundaries, we show that
S_{p,q} is homeomorphic to S_{p',q'} if and only if {d,e}={d',e'}. In
fact, we will show a topological construction of S_{p,q} via a 0-surgery
on a 2-bridge knot. Eventually, coming back to algebraic geometry, we
will use a variant of logMMP to show that surfaces S_{p,q} exhaust all
affine surfaces of Kodaira dimension zero, whose coordinate rings are
factorial and have trivial units. This is a joint work with P. Raźny.