Higher rank GKZ-fans are defined for point configurations and used to construct quasi-valuations inducing flat degenerations of toric varieties to reduced unions that encode the associated polytopal subdivisions.
Rees,Lectures on the Asymptotic Theory of Ideals, London Mathematical Society Lecture Note Series.113, Cambridge University Press (1988)
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Higher rank Gelfand-Kapranov-Zelevinsky fans
Higher rank GKZ-fans are defined for point configurations and used to construct quasi-valuations inducing flat degenerations of toric varieties to reduced unions that encode the associated polytopal subdivisions.