The Hesse curve of a Lefschtz pencil of plane curves

We prove that for a generic Lefschetz pencil of plane curves of degree $d\geq 3$thereexistsacurve$H$(calledtheHessecurveofthepencil)ofdegree$6(d-1)$andgenus$3(4d^2-13d+8)+1$,andsuchthat:$(i)$ $H$ 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.