For a complete, smooth toric variety Y , we describe the graded vector space T 1 Y . Furthermore, we show that smooth toric surfaces are unobstructed and that a smooth toric surface is rigid if and only if it is Fano. For a given toric surface we then construct homogeneous deformations by means of Minkowski decompositions of polyhedral subdivisions, compute their images under the Kodaira-Spence...

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism...

The purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that ...

PURPOSE To compare lens orientation and rotational recovery of five currently available soft toric lenses. METHODS Twenty subjects were recruited and trialed with each of the study lenses in a random order. Study lenses were PureVision(®) Toric (B&L), Air Optix(®) for Astigmatism (Alcon), Biofinity(®) Toric (CooperVision), Acuvue(®) Advance for Astigmatism (Vistakon), and Proclear(®) Toric (C...

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,R)-equivalence and deduce that the only gkz-rational toric four-folds in P6 are those varieties associated with an essential Cayley configuration....

PURPOSE To compare the astigmatic power of toric intraocular lenses (IOLs) obtained from the AcrySof, TECNIS, and iTrace toric calculator in patients with preoperative with-the-rule (WTR) or against-the-rule (ATR) corneal astigmatism. MATERIALS AND METHODS Fifty eyes with cataract and corneal astigmatism greater than 0.75 diopters were enrolled in each group (WTR and ATR). Keratometric values...

We consider the action of a subtorus of the big torus on a toric variety. The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety. Moreover, we study the relations of such quotients with good quot...

Obviously, toric lenses have to remain in stable positions after implantation bearing in mind that for every 10 degrees of misalignment, there is one third loss of toric correction. While some time ago this was still a problem, toric IOLs of today are significantly less affected by lens rotation. Another disturbing issue with toric lenses has also been fixed in the meantime: according to a new ...

We completely classify toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms.

Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck K-theory of simplicial toric stack bundles and study the Chern character homomorphism.