Abstract. In this paper, we discuss the relative K-stability and the modified K-energy associated to the Calabi’s extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative Kstability and the properness of modified K-energy. In particular, our results hold for toric Fano manifolds with vanishing Futaki-...

Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an arbitrary polynomial algebra at the algebraic torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope. We stud...

Examples 1.1. • Y ∼= P, D a nodal cubic. • Y a complete nonsingular toric surface, D the toric boundary (i.e, the complement of the big torus orbit Gm). • Blowups: Given a Looijenga pair (Y ,D), we can take “toric blowups,” where Y is the blowup of Y at a nodal point of D and D is the inverse image of D). Note that toric blowups do not change U . “non-toric blowups,” where Ỹ is the blowup of Y ...

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the corr...

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel’fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeom...

These notes survey some basic results in toric varieties over a field F , with examples and applications. A computer algebra package (written by the author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both MAGMA [MAGMA] and GAP [GAP]). Among other things, the package implements the desingularization procedure, constructs some ...

Let IM and IN be defining ideals of toric varieties such that IM is a projection of IN , i.e. IN ⊆ IM . We give necessary and sufficient conditions for the equality IM = rad(IN + (f1, . . . , fs)), where f1, . . . , fs belong to IM . Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetica...

RESULTS: Implantation of a toric IOL reduced the refraction astigmatism by 80%. Aberrations increased in all patients, but while astigmatic aberration decreased with toric IOLs, spherical aberration decreased with aspheric IOLs. The UCVA was better with toric and aspheric IOLs (0.1 logMAR), whereas the BCVA was statistically better with aspheric IOLs compared to spherical (p = 0.05). The CSF wa...

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then ...

Abstract: Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of...