We introduce and study the Hesse pencil variety \(H_8\) , obtained as the Zariski closure in the Grassmannian G(1, 9) of the set of pencils generated by a smooth plane cubic and its Hessian. We prove that \(H_8\) has dimension 8 and can be realized as the intersection of G(1, 9) with ten hyperplanes corresponding to the Schur module \(\mathbb {S}_{(5,1)}\mathbb {C}^3\) . Moreover, \(H_8\) coincides with the closure of the special linear group SL(3)-orbit of the pencil \(\langle x^3+y^3+z^3,\ xyz\rangle \) and contains eight additional orbits. The variety is singular, and its singular locus is precisely the union of two orbits, \(O(\langle x^3,x^2y\rangle )\) and \(O(\langle x^2y,x^2z\rangle )\) . A key ingredient in our study is a cubic skew-invariant \(R \in \bigwedge ^3(\textrm{Sym}^3\mathbb {C}^3)\) , defined by \(R(l^3,m^3,n^3) = (l \wedge m \wedge n)^3\) , where l, m, n are linear forms in \((\mathbb {C}^3)^*\) . The vanishing of R characterizes pencils generated by a cubic and its Hessian, and it allows us to write explicit equations defining \(H_8\) . A crucial geometric step in our argument is the fact that through four general points of \(\mathbb {P}^2\) there pass exactly six Hesse configurations, which enables us to compute the multidegree of \(H_8\) and conclude that it coincides with the variety defined by the invariant R.
