Algebraic Geometry Seminar
Alfred Chen
National Taiwan University
Birational geometry of threefolds of general type
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.
Thursday December 6, 2007 at 4:00 PM in SEO 636