Study on integral convex polytopes and toric varieties

绪方研究理想定义投射环面品种。我们说理想满足性质（Np），如果它的自由分解是最简单的，直到次数p部分。我们得到了这个p关于簇的维数和同一样本线丛的张量积的次数的一个可能的最佳估计。我们还得到了Fakhruddin定理的一个代数几何证明，该定理指出：Ample丛的任何整体截面乘以nef丛都是Ample丛和nef丛的截面的倍数。其次，证明了非奇异环面3-fold上的一个充分线丛是正规生成的，如果这个丛加上规范丛的两倍后没有整体截面。Ishida利用Zerkiki-Riemann空间的概念研究了真实的fans的完备化。我们还确定了由Hirotaka Ishida定义的CCI曲面的模数，Hara研究了正特征的二维孤立奇点。我们定义了一对曲面和一个有效因子的F-pure阈值ftp（X，D），它是对数正则阈值的模拟。Kajiwara研究了热带几何与复曲面几何之间的关系，并利用p-分形方法证明了ftp（X，D）的值为有理数。我们刻画了由热带超曲面自然决定的多面体分解所定义的投射环面簇的退化，并从代数几何的角度重建了热带环面簇.作为结果，我们得到我们还重建热带环面品种的代数几何。作为结果，我们得到了热带非奇异复曲面的交理论。

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.