Alfred Chen : Birational geometry of threefolds of general type

Posted by , part of the Algebraic Geometry Seminar.

At

Dec. 6, 2007, 4 p.m.

In

SEO 636

Abstract

Given a variety $X$ of general type, by definition, the $m$- canonical map is birationally stable for $m$ sufficiently large. For curves, it is a classical result that $m$-canonical map is an embedding for $m \ge 3 $. For surfaces, it's known that $m$-canonical map is birational for $m \ge 5 $. Only very recently, it is proved that there is a constant $c(n)$ depending only on $\dim X$ such that $m$-canonical map is birational for all $m \ge c(n)$. In a recent joint wotk with Meng Chen, we found an explicit bound for $c(3)$. More precisely, we are able to prove that for any complex projective threefold of general type $X$, one has: 1. $Vol(X) \ge 1/2660$, 2. $P_{12}(X) >0$, 3. $P_{24}(X) \ge 2$, 4. $m$-canonical map is birational for all $m \ge 77$. We are going to show some more appplications of our method.