Joey Lee : Weak Northcott property and the D-ratio for rational maps

Posted by Laura DeMarco , part of the Number Theory Seminar.

At

Sept. 8, 2010, 3 p.m.

In

SEO 612

Abstract

In 1950, Northcott proved the height inequality for morphisms on projective spaces. Let $f:P^n\to P^n$ be a morphism. Then, there is a constant $C$ depending on the given morphism such that the height function $h$ satisfies $h(f(P)) + C > \deg f \, h(P) > h(f(P)) - C$ for all points $P \in P^n$. Unfortunately, the upper bound of the inequality does not hold for rational maps. However, we can find weaker inequality by 1) restricting points on an affine open set and 2) enlarging the upper bound a little bit. In this talk, I will introduce the D-ratio for a rational map on projective space and prove the Weak Northcott property for rational maps.

This seminar is joint with Geometry/Topology/Dynamics.