A History of Flips in Combinatorial Triangulations

Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of the various attempts to answer it.Comment: Added a paragraph referencing earlier work in the vertex-labelled setting that has implications for the unlabeled settin

Non-connected toric Hilbert schemes

We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc., 13:3 (2000), 611--637 not only in size, but also in that their toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.Comment: 18 pages, 2 figures. Except for Remark 2.6 (see below) changes w.r.t. version 2 are mostly minor editings suggested by an anonimous referee of "Mathematische Annalen". The paper has been accepted in that journal. Most of the contents of Remark 2.6 have been deleted, since there was a flaw in the argumen