5 Some Geometry and Combinatorics for the S - Invariant of Ternary

In earlier papers [5,4], the S-invariant of a ternary cubic f was related to the curvature of the level set f = 1 in R 3. In particular, when f arises from the cubic form on the second cohomology of a smooth projective threefold with second Betti number three, the value of the S-invariant is closely linked to the behaviour of this curvature on the open subset of this level set consisting of Kähler classes [5]. In this paper, we consider the cubic forms arising from complete intersections in the product of three projective spaces, and investigate various conjectures of a combinatorial nature suggested concerning their invariants.