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

Abstract

In earlier papers, P. M. H. Wilson, “Sectional Curvatures of Kähler Moduli,” and B. Totaro, “The Curvature of a Hessian Metric,” 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 3 of this paper. In the case that $f$ 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 behavior 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.