Higher rank Gelfand-Kapranov-Zelevinsky fans

Abstract

We define and study the higher rank GKZ-fans of point configurations, serving as the set of discrete and homogeneous quasi-valuations on the homogeneous coordinate ring of the associated toric variety, where the rank one cases coincide with the usual GKZ-fans. Such a quasi-valuation is then used to degenerate the toric variety flatly to a reduced union of toric varieties, which encodes the polytopal subdivision arising from the point in the higher rank GKZ-fan.

27 pages, v2: the v1 of the preprint has been split into two preprints: here is the first part concentrating on higher rank GKZ-fans; a second preprint will be prepared according to the remaining part of v1. Comments are welcome