Lefschetz pencils on a complex projective plane from a topological viewpoint

Abstract

In this article, we present a differential topological construction of symplectic Lefschetz pencils of genus $\frac{(d-1)(d-2)}{2}$ with $d^2$ base points and $3(d-1)^2$ critical points for arbitrary $d\geq 4$, analogous to the holomorphic Lefschetz pencils of curves of degree $d$ in $\mathbb{C}P^2$. Moreover, for the case $d=4$, we derive an explicit monodromy factorization of the genus $3$ holomorphic Lefschetz pencil on $\mathbb{C}P^2$ based on the braid monodromy technique and prove that it can also be topologically constructed by breeding the monodromy relations of the genus $1$ holomorphic Lefschetz pencils.