# The Hesse curve of a Lefschtz pencil of plane curves

## TL;DR

This paper introduces the Hesse curve associated with a generic Lefschetz pencil of plane curves, detailing its degree, genus, singularities, and its relation to inflection points and base points of the pencil.

## Contribution

It establishes the existence and properties of the Hesse curve for a Lefschetz pencil of plane curves of degree d ≥ 3, including its degree, genus, and singularity structure.

## Key findings

- Hesse curve has degree 6(d-1) and specific genus.
- H has d^2 singular points of multiplicity three at base points.
- H intersects each pencil member at inflection points and base points.

## Abstract

We prove that for a generic Lefschetz pencil of plane curves of degree $d\geq 3$ there exists a curve $H$ (called the Hesse curve of the pencil) of degree $6(d-1)$ and genus $3(4d^2-13d+8)+1$, and such that: $(i)$ $H$ has $d^2$ singular points of multiplicity three at the base points of the pencil and $3(d-1)^2$ ordinary nodes at the singular points of the degenerate members of the pencil; $(ii)$ for each member of the pencil the intersection of $H$ with this fibre consists of the inflection points of this member and the base points of the pencil.