Algebraic Geometry Seminar
Mike Roth
Queen's University
Seshadri constants, diophantine approximation, and Roth's theorem for arbitrary varieties
Abstract: If $X$ is a variety of general type defined over a number field $k$, then the
Bombieri-Lang conjecture predicts that the $k$-rational points of $X$ are not
Zariski dense. One way to view the conjecture is that a global condition on
the canonical bundle (that it is ''generically positive'') implies a global
condition about rational points. By a well-established principle in
geometry we should also look for local influence of positivity on the
accumulation of rational points. To do that we need measures of both these
local phenomena.
Let $L$ be an ample line bundle on $X$, and $x\in X(\overline{k})$. By
slightly modifying the usual definition of approximation exponent on
$\mathbf{P}^1$, we define a new invariant $\alpha_{x}(L)\in (0,\infty]$ which
measures how quickly rational points accumulate around $x$, as measured by
$L$.
The central theme of the talk is the interrelations between $\alpha_x(L)$ and
the Seshadri constant $\epsilon_{x}(L)$ which measures the local positivity
of $L$ near $x$. In particular, the classic approximation theorem of Klaus
Roth on $\mathbf{P}^1$ generalizes as an inequality between $\alpha_{x}$ and
$\epsilon_{x}$ valid for all projective varieties. This is joint work with
David McKinnon.
Wednesday October 17, 2012 at 4:00 PM in SEO 427