  Given a real cubic form f(x,y,z), there is a pseudo-Riemannian metric given
by its Hessian matrix, defined on the open subset of R^3 where the Hessian
determinant h is non-zero. We determine the full curvature tensor of this
metric in terms of h and the S-invariant of f, obtaining in the process various
different characterizations of S. Motivated by the case of intersection forms
associated with complete intersection threefolds in the product of three
projective spaces, we then study ternary cubic forms which arise as follows: we
choose positive integers d1, d2, d3, set r = d1 + d2 + d3 - 3, and consider the
coefficient F(x,y,z) of H1^d1 H2^d2 H3^d3 in the product (x H1 + y H2 + z H3)^3
(a_1 H1 + b_1 H2 + c_1 H3) ... (a_r H1 + b_r H2 + c_r H3), the a_j, b_j and c_j
denoting non-negative real numbers; we assume also that F is non-degenerate.
Previous work of the author on sectional curvatures of Kahler moduli suggests a
number of combinatorial conjectures concerning the invariants of F. It is
proved here for instance that the Hessian determinant, considered as a
polynomial in x,y,z and the a_j, b_j, c_j, has only positive coefficients. The
same property is also conjectured to hold for the S-invariant; the evidence and
background to this conjecture is explained in detail in the paper.
