# Algebraic Geometry Seminar

#### 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.