# Mathematics Colloquium

#### Fibrations in Rational Surfaces and Their Sections

**Speaker:**
Brendan Hassett, Rice

**Location:**
Warren Weaver Hall 1302

**Date:**
Monday, February 23, 2015, 3:45 p.m.

**Synopsis:**

A central problem of Diophantine geometry is to formulate effective criteria for the existence of solutions to polynomial equations over various fields. While this is intractible in general, the case of rational surfaces over global fields has precise conjectures supported by many specific examples. We focus on the geometric case of function fields of curves, where solutions may be interpreted as sections of a fibration. Here we classify the fibrations in geometric terms and employ this to deduce concrete (and sometimes surprising) implications of the Diophantine conjectures. (joint with Kresch and Tschinkel)