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

In earlier papers [Wilson 04, Totaro 04], the S-invariant of a ternary cubic f was interpreted in terms of the curvature of related Riemannian and pseudo-Riemannian metrics — this is clarified further in Section 1. In the case 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 cone consisting of Kähler classes. In this paper, we concentrate on the cubic forms arising from complete intersection threefolds in the product of three projective spaces, and investigate various conjectures of a combinatorial nature arising from their invariants.