Noriyuki Hamada
An example of genus-$2$ smallest Lefschetz fibration and pencil
An example of genus-$2$ Lefschetz fibration over the $2$-sphere having
the minimal number of critical points (which is $7$) was given by Xiao
in an algebro-geometric manner in the 1980s, but its monodromy
representation was not explicit.
It had been the only known example utill recently Baykur-Korkmaz finally
discovered an explicit monodromy factorization that gives a smooth
Lefschetz fibration of Xiao's type.
In this talk, we review their construction and simplify its vanishing
cycles.
Furthermore, we give a set of three disjoint $(-1)$-sections of that
fibration, which in turn yields a minimal Lefschetz pencil.
Some applications will be also discussed.
This is a joint work with R. I. Baykur (University of Massachusetts) and
M. Korkmaz (Middle East Technical University).