ODTÜ-BİLKENT
ALGEBRAIC GEOMETRY SEMINAR

“PRODUCTS OF TWO DEHN TWISTS AND REAL LEFSCHETZ
FIBRATION”

by

ALEXANDER DEGTYAREV

Abstract: An object repeatedly occurring in algebraic geometry is
a fibration with singular fibers. If the base is a topological disk and the
number of singular fibers is finite, the topology of such a fibration can
adequately be described by the so called monodromy factorization of
the monodromy at infinity (boundary of the disk), regarded up to a
certain collection of moves, called Hurwitz moves and, possibly,
global conjugation. We consider the simplest nontrivial case: factorizations
into product of two Artin generators in the braid group B_3 (equivalently, two
Dehn twists in the mapping class group of a torus). Even here, the results
obtained are quite unexpected; considering the known examples, the general case
(even in B_3) is very far from its complete understanding. Trivial as it seems,
this simplest case has a number of geometric applications. As a first one, we
prove that any maximal real elliptic Lefschetz fibration over the sphere is
algebraic. Other applications include the
semi-simplicity statement for real trigonal M-curves in Hirzebruch
surfaces. (One may try to speculate that products of two Dehn twists
are still `tame' precisely because they are related to maximal geometric
objects.) The principal tool is a description of subgroups of the modular group
in terms of a certain class of Grothendieck's dessins d'enfants, followed by high school geometry. (joint with Nermin Salepci, Université de Lyon)

Date:  7 October  2011 Friday

Time: 15.40

Place: Bilkent,
Mathematics Seminar Room SA-141

Tea and cookies will be served after the
talk.