Linear algebra, combinatorics and defectivity

A DICKENSTEIN

Seattle

Congreso; CCAAGS 22; 2022

In joint work with E.Cattani, we explore four approaches to the question of defectivity for acomplex projective toric variety associated with an integral configuration A.The first one arises from the explicit tropicalization of the associated dualvariety given with E.M. Feichtner and B. Sturmfels. The second one is based onthe notion of iterated circuits introduced by A. Esterov. The third one uses aninvariant defined by R. Curran and E. Cattani in terms of a Gale dualconfiguration of A. The fourth one was proposed by K. Furukawa and A. Ito interms of Cayley decompositions of A.We obtain formulae forthe dual defect in terms of iterated circuits and Gale duals. Our proofs andextended definitions are linear-algebraic in nature and hold for finite configurationsover an arbitrary field of characteristic zero. We prove in this generality theequivalence of the first three approaches. We give a Gale dual interpretationof Cayley decompositions and apply it to the study of defective configurations.However, we only prove one inequality between the first three invariants andthe fourth one. I will close my talk with an open question concerning thereverse inequality.