Classification of Genus-1 Holomorphic Lefschetz Pencils

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.