Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration

Abstract

We give a maximal set of disjoint (−1)-sections of the well-known Lefschetz fibration constructed by Matsumoto, Cadavid and Korkmaz. In fact, we obtain several such sets for a fixed genus, which implies that the Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils. We also determine the diffeomorphism types of the obtained supporting minimal Lefschetz pencils.

Classification of genus-$1$ holomorphic Lefschetz pencils

“…For a Lefschetz pencil f , the genus of the closure of a regular fiber is called the genus of f . a The first author also found factorizations in [9,10] in different contexts, which in fact can be shown to be Hurwitz equivalent to either Korkmaz-Ozbagci's or Tanaka's.…”

“…Although the isomorphism classes of the pencils f n and f s do not depend on the choice of generic projections P 9 P 1 (cf. remark 2.2), one cannot deduce immediately from theorem 1.1 that the isomorphism class of a blow-up of f n does not depend on the choice of blown-up base points (one can indeed find in [10] examples of a pair of non-isomorphic pencils that are obtained by blowing-up a common pencil at the same number but different combinations of base points). We next address this issue by examining the monodromies of f n and f s .…”