Experimental Mathematics

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

P. M. H. Wilson

Full-text: Open access

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.

Article information

Source
Experiment. Math., Volume 15, Issue 4 (2006), 479-490.

Dates
First available in Project Euclid: 5 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.em/1175789782

Mathematical Reviews number (MathSciNet)
MR2293598

Zentralblatt MATH identifier
1172.14332

Subjects
Primary: 15A72: Vector and tensor algebra, theory of invariants [See also 13A50, 14L24]
Secondary: 32J27: Compact Kähler manifolds: generalizations, classification 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx] 53A15: Affine differential geometry

Keywords
Ternary cubics invariant theory curvature combinatorial inequalities

Citation

Wilson, P. M. H. Some Geometry and Combinatorics for the $S$-Invariant of Ternary Cubics. Experiment. Math. 15 (2006), no. 4, 479--490. https://projecteuclid.org/euclid.em/1175789782


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