Orenall Study on Algebraic Varieties and its Applications

katsura主要研究了代数几何的正特征与编码理论之间的关系。他得到了完全孤立和孤立半径的概念。他证明了[7,4,3]-汉明码是完全隔离的，其隔离半径等于2ROO<2>/77。他还和J.katsuta一起检查了Golay代码，几乎完成了他们的计算。藤田（T.Fujita）将Kodaira能量小于1/2的偏振三倍分类。Y.Kawamata在充足的线束上处理了Fujita的猜想，并在三维和四维情况下得到了肯定的答案。他还研究了Calabi-Yau流形的除数组成的锥，并证明了具有正Kadaira维数的三折体存在最多有限多个极小模型。M.Maruyama研究了半稳定向量束的模空间，并证明了实例的模空间的连通性。他还成功地构造了稳定轮系的模空间。M.Miyanishi得到了Roberts对Hilbert第十四问题的反例的简单证明，也得到了关于开放代数变分的一些结果。S.Mori考虑了平面群方案在代数空间上的作用，得到了商的存在性的一般结果。Y.Morita研究了代数曲面的有理点，得到了关于Batirev-Manin猜想的一些结果。T.Shioda研究了Mordell-Weil格，作为应用，他得到了大秩有理数域上的2属代数曲线。Y.Sumihiro研究了向量束的基本变换和与向量束相关的行列式流形，提出了一种研究射影空间上向量束的新方法。

T.Katsura mainly studied the relation between algebraic geometry in positive characteristic and coding theory. He got notions of complete isolatedness and isolated radius. He showed that the [7,4,3]-hamming code is completely isolated and that its isolated radius is equal to 2ROO<2>/77. He also examined the Golay code with J.katsuta and almost finished their calculation. T.Fujita classified the polarized threefolds with Kodaira energy less than-1/2. Y.Kawamata treated Fujita's conjecture on ample line bundles and got the affirmative answer to it for three and four dimensional cases. He also investigated the cone consisted of divisors of a Calabi-Yau manifold, and showed that there exist at most finitely many minimal models for threefolds with positive Kadaira dimension. M.Maruyama studied the moduli space of semi-stable vector bundles, and showed the connectedness of the moduli space of instantons. He also succeeded in the construction of moduli space of stable sheaves. M.Miyanishi got a simple proof of the counter example by Roberts to the fourteenth problem of Hilbert, and also got some results on open algebraic varieties. S.Mori considered the action of flate group schemes on algebraic spaces, and got a general result on the existence of the quotients. Y.Morita studied rational points of algebraic surfaces and got some results on Batirev-Manin's conjecture. T.Shioda studied Mordell-Weil lattices, and as an application, he got algebraic curves of genus 2 over the rational number field with big rank. Y.Sumihiro studied the fundamental transformation of vector bundle and the determinant manifold associated to a vector bundle, and proposed a new method of the study of vector bundles on projective spaces.