The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is ...

The main purpose of this paper is to give a simple and non-combinatorial proof of the toric Mori theory. Here, the toric Mori theory means the (log) Minimal Model Program (MMP, for short) for toric varieties. We minimize the arguments on fans and their decompositions. We recommend this paper to the following people: (A) those who are uncomfortable with manipulating fans and their decompositions...

We treat a generalization of Danilov’s vanishing theorem on toric polyhedra. A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We also give a proof of the E1-degeneration of Hodge to de Rham type spectral sequence for toric polyhedra in any characteristic.

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is ...

Correction of corneal astigmatism is a key element of cataract surgery, since post-surgical residual astigmatism can compromise the patient's uncorrected visual acuity. Toric intraocular lenses (IOLs) compensate for corneal astigmatism at the time of surgery, correcting ocular astigmatism. They are a predictable treatment. However, accurate measurement of corneal astigmatism is mandatory for ch...

ince its adoption in 1997, the Unified Modeling Language (UML) has proved very popular with software engineers and has become the de facto standard as a visual modeling language for software engineers. However, this software focus of UML has discouraged many systems engineers from adopting it in earnest. Those who did adopt UML developed strategies to cope with its shortcomings. A common approa...

This note proves combinatorially that the intersection pairing on the middle-dimensional compactly supported cohomology of a toric hyperkähler variety is always definite, providing a large number of non-trivial L2 harmonic forms for toric hyperkähler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L2 harmonic forms on complete hyperk...

2 Toric Varieties 7 2.1 Constructing an atlas for a toric variety from a fan . . . . . . . . . . . . . . . . . . . . 7 2.2 Cones correspond to torus orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Maps of fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We characterize the orientability of an abstract real toric variety as well as the orientability of a toric subvariety of a sphere. We also determine the number of components of the smooth locus of a toric variety. These results are proven for an extension of the Davis-Januskiewicz notion of a small cover to singular spaces. We characterize the orientability of toric varieties associated to pos...

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for se...