Algebraic Geometry Seminar

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Almost all plane curves are simply connected

Speaker: Michael McQuillan, IHES

Location: Warren Weaver Hall 317

Date: Tuesday, October 3, 2017, 3:30 p.m.

Synopsis:

Obviously the title is false, but over
Spec Z it might be true, albeit that a moments
thought reveals that the complex fibre should be
uni-branch. This is, however, just an example of
the principle observed by Jean-Benoit Bost that
if one replaces geometric dimension by absolute
dimension then Lefschetz theorems can still hold
in the presence of suitable "arithmetic ampleness".
In the  specific, Bost made an arithmetic version
of Castelnuovo's numerical proof that an ample
divisor in a normal surface is connected, but the
theory didn't really develop because of the lack
of a proof of geometric Lefschetz for higher
homotopy groups that admits the possibility of being
arithmeticised. Regular seminar goers may, however,
recall that a couple of years ago I gave a new proof
of geometric Lefschetz that works in vast generality,
and since then arithmeticising it has been a project
with Federico Buonerba, who gave a progress report
in the seminar last year. The current status is that
the manuscript might even be finished before the
talk, and the results, including the claim of the
title, are better than expected.