Study of Moduli and its Applications

得到了如下结果：1)射影平面上第二类稳定向量丛的跳线曲线深刻地反映了向量丛的结构。我们找到了一种确定曲线奇点上的重数和切锥的方法。2)我们证明了曲面上的普通双点与其上自反叶的变形空间之间的密切关系。3)我们推广了Bogomolov不等式，并发现了与稳定曲线的模空间的一个有趣的关系。4)研究了退化变种上的稳定叶，给出了通过一个更简单的变量的组合来寻找非奇异簇上稳定叶的模空间结构的方法。5)我们构造了具有这种极化的曲面族稳定向量丛的模空间的维度比预期的要大，即使对于很大的第二类陈氏也是如此。6)期待进一步发展的课题之一是复层复数的模空间。为了得到好的模，我们必须给出稳定复的适当定义。相反，我们引入了简单复数的概念，并在代数空间范畴中构造了它们的模空间。7)半稳定层在正特征情形下的有界性是最重要的问题之一。事实上，这一直是该项目要解决的主要目标。Maruyama在1973年证明了曲面的有界性，并在1980年证明了第二、三阶情形的有界性。经过20年的没有进展，我们可以向前迈进一步，就是成功地证明了排名4的案件。遗憾的是，我们所采用的方法不能直接推广到高阶情形，我们必须寻找另一种方法来完成对一般情形的证明。

The results we have obtained are the following.1) The curve of the jumping lines of second kind of a stable vector bundle of rank 2 on the projective plane deeply reflects the structure of the vector bundle. We found a way to determine the multiplicity and the tangent cone at a singular pint of the curve.2) We showed a close relationship between an ordinary double point on a surface and the space of deformations of a reflexive sheaf on it.3) We generalized Bogomolov's inequality and found an interesting relationship with the moduli space of stable curves.4) Stable sheaves on a degenerating family of varieties were studied and gave a way to find the structure of the moduli space of stable sheaves on a nonsingular variety through a combination of simpler varieties.5) We constructed families of surfaces on which there is a polarization such that the dimension of the moduli space of stable vector bundles is bigger than expected even for very big second Chern classes.6) One of the subjects expected further development is the moduli space of complexes of sheaves. To get good moduli we have to give a proper definition of stable complex. Instead of this we introduced the notion of simple complex and constructed their moduli space in the category of algebraic spaces.7) One of the most important problem is the boundedness of semistable sheaves in positive characteristic cases. In fact, this has been the main target to be solved in this project. The boundedness for a surface was proved in 1973 and for the rank 2, 3 cases in 1980 by M.Maruyama. After 20 years of no progress, we could take a step forward, that is, succeeded in proving the rank 4 case. Unfortunately the method we took cannot be generalized to higher rank cases directly and we nay have to find another way to complete the proof for general cases.