  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.
