Classification of Genus-1 Holomorphic Lefschetz Pencils

Noriyuki Hamada, Kenta Hayano

Research output: Contribution to journalArticlepeer-review


In this paper, we classify relatively minimal genus-1 holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on P1× P1of bidegree (2, 2) or a blow-up of the pencil on P2of degree 3, provided that no fiber of a pencil contains an embedded sphere (note that one can easily classify genus-1 Lefschetz pencils with an embedded sphere in a fiber). We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blow-up of the pencil on P2of degree 3 does not depend on the choice of blown-up base points. We also show that the genus-1 Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine base points) and Tanaka (with eight base points) are respectively isomorphic to the pencils on P2and P1× P1above, in particular these are both holomorphic.

Original languageEnglish
Pages (from-to)1079-1119
Number of pages41
JournalTurkish Journal of Mathematics
Issue number3
Publication statusPublished - 2021


  • Lefschetz pencil
  • braid monodromy
  • holed torus relation
  • monodromy factorization

ASJC Scopus subject areas

  • General Mathematics


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