Some geometry and combinatorics for the \(S\)-invariant of ternary cubics. (English) Zbl 1172.14332

Wilson, P. M. H.

Exp. Math. 15, No. 4, 479-490 (2006).

Summary: In earlier papers [P. M. H. Wilson, Math. Ann. 330, No. 4, 631–664 (2004; Zbl 1067.32015); B. Totaro, Int. J. Math. 15, No. 4, 369–391 (2004; Zbl 1058.53032)], 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.

MSC:

14J40	\(n\)-folds (\(n>4\))
14J32	Calabi-Yau manifolds (algebro-geometric aspects)
32Q25	Calabi-Yau theory (complex-analytic aspects)

Keywords:

Ternary cubics; invariant theory; curvature; combinatorial inequalities

Citations:

Zbl 1067.32015; Zbl 1058.53032