The main purpose of this notes is to supplement the paper [Re], which treated Minimal Model Program (also called Mori’s Program) on toric varieties. We calculate lengths of negative extremal rays of toric varieties. As an application, we obtain a generalization of Fujita’s conjecture for singular toric varieties. We also prove that every toric variety has a small projective toric Q-factorializa...

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of...

The moduli spaceM0,n plays important roles in algebraic geometry and theoretical physics. Yet, some basic properties of M 0,n still remain open. For example, M0,n is rational and nearly toric (that is, it contains a toric variety as a Zariski open subset), but it is not a toric variety itself starting from dimension 2 (n ≥ 5). So, a basic question is: Can it be degenerated flatly to a projectiv...

1. Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Characters and 1-Parameter Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3. Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . ....

PURPOSE To prospectively evaluate postoperative clinical outcomes with implantation of toric intraocular lenses (IOLs) using preoperative keratometry from an optical low coherence reflectometry (OLCR) ocular biometer (Lenstar® LS900) and the built-in Barrett toric calculator. PATIENTS AND METHODS A prospective observational study recruited one or both eyes of subjects who underwent uncomplica...

PURPOSE To determine whether rotational stability is improved with the longer Staar toric intraocular lens (IOL). SETTING A private practice, Los Altos, California, USA. METHODS Staar Surgical manufactures toric plate-haptic IOLs under +24.0 diopters (D) in 2 lengths. Fifty consecutive eyes requiring toric IOL spherical powers less than +24.0 D received the longer toric IOL (TL; 11.2 mm) us...

Extending the notion of indispensable binomials of a toric ideal ([14], [7]), we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for proving the existence or non-existence of a unique minimal system of binomials generators of a toric ideal. Some examples of indispensable monomials ...

We construct one-parameter complex analytic families whose special fibers are complete toric varieties. Under some assumptions, the general fibers of these families also become toric varieties and we can explicitly describe the corresponding fans from the data of the fans corresponding to the special fibers. Using these families, we give a deformation family for a certain toric weakened Fano 3-...

The GIT chamber decomposition arising from a subtorus action on a polarized quasiprojective toric variety is a polyhedral complex. Denote by Σ the fan that is the cone over the polyhedral complex. In this paper we show that the toric variety defined by the fan Σ is the normalization of the toric Chow quotient of a closely related affine toric variety by a complementary torus.