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

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.