Non-splitting flags, Iterated Circuits, $\underline{\mathbf \sigma}$-matrices and Cayley configurations

TL;DR

This paper investigates the defectivity of complex projective toric varieties using four approaches, linking combinatorial invariants, Gale duals, and Cayley decompositions to provide new formulas for dual defect.

Contribution

It introduces new formulas for the dual defect of toric varieties based on affine invariants, Gale duals, and Cayley decompositions, connecting multiple combinatorial approaches.

Findings

Derived formulas for dual defect using affine invariants and Gale duals.

Linked Cayley decompositions to defectivity via Gale dual interpretation.

Provided computational methods for defectivity in toric varieties.

Abstract

We explore four approaches to the question of defectivity for a complex projective toric variety $X_A$ associated with an integral configuration $A$. The explicit tropicalization of the dual variety $X_A^\vee$ due to Dickenstein, Feichtner, and Sturmfels allows for the computation of the defect in terms of an affine combinatorial invariant $\rho(A)$. We express $\rho(A)$ in terms of affine invariants $\iota(A)$ associated to Esterov's iterated circuits and $\lambda(A)$, an invariant defined by Curran and Cattani in terms of a Gale dual of $A$. Thus we obtain formulae for the dual defect in terms of iterated circuits and Gale duals. An alternative expression for the dual defect of $X_A$ is given by Furukawa-Ito in terms of Cayley decompositions of $A$. We give a Gale dual interpretation of these decompositions and apply it to the study of defective configurations.