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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 XA 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 ρ(A). We express ρ(A) in terms of affine invariants ι(A) associated to Esterov’s iterated circuits and λ(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 XA 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.

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Acknowledgements

A. Dickenstein was partially supported by UBACYT 20020100100242, CONICET PIP 20110100580, and ANPCyT PICT 2013-1110, Argentina.

We are indebted to Atsushi Ito for fruitful discussions and for allowing us to quote his counterexample 3. We thank the referees for their detailed and useful comments to improve our manuscript. Almost 30 years ago, at an NSF Geometry Institute held at Amherst College, Bernd Sturmfels introduced us to many of the topics discussed here. We are grateful for his inspiration and his friendship throughout this time.

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Correspondence to Eduardo Cattani.

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Dedicated to Bernd Sturmfels on the occasion of his 60th birthday.

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Cattani, E., Dickenstein, A. Non-splitting Flags, Iterated Circuits, \(\underline {\mathbf {\sigma }}\)-Matrices and Cayley Configurations. Vietnam J. Math. 50, 679–706 (2022). https://doi.org/10.1007/s10013-022-00554-7

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  • DOI: https://doi.org/10.1007/s10013-022-00554-7

Keywords

  • Projective toric varieties
  • Dual defect
  • Cayley decompositions
  • Gale dual
  • Tropical variety
  • Iterated circuits
  • Non-splitting flags

Mathematics Subject Classification (2010)

  • Primary 14M25
  • Secondary 05E14; 14T15