Study on integral convex polytopes and toric varieties

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Ogata studied about ideals defining projective toric varieties. We say the ideal satisfies the property (Np) if its free resolution is the simplest up to the degree p part. We obtained a best possible estimate of this p with respect to the dimension of the variety and the times of tensor product of the same ample line bundle. We also obtained an algebro-geometric proof of the Theorem of Fakhruddin, which states that any global section of Ample bundle multiplied by a nef bundle is a nultiple of sections of the ample bundle and the nef bundle. Next we proved that an ample line bundle on a nonsingular toric 3-fold is normally generated if the bundle after added with the twice of the canonical bundle has no global sections. As a consequence, we showed that the anti-canonical bundle on nonsingular toric Fano 4-fold is normally generated.Ishida studied about completion of real fans by using the notion of Zariski-Riemann spaces. We also determined the moduli number of CCI surfaces which are defined by Hirotaka Ishida.Hara studied about isolated singularities of dimension two in positive characteristic. We define the F-pure threshold ftp(X, D) for a pair of a surface and an effective divisor which is an analog of the logarithmic canonical threshold. And we showed that ftp(X, D) has values in rational numbers by using the method of p-fractal.Kajiwara studied abuot the relation between tropical geometry and toric geometry. We characterized degenerations of projective toric varieties defined from polyhedral decompositions which are determined naturally by tropical hypersurfaces.We also reconstructed tropical toric varieties in terms of algebraic geometry. As a consequence, we obtainedWe also reconstructed tropical toric varieties in terms of algebraic geometry. As a consequence, we obtained an intersection theory on tropical nonsingular toric surfaces.