Free resolutions of the coordinate rings of projective varieties

我的研究一直致力于研究射影簇的坐标环的极小自由分解。特别是，Castelnuovo-Mumford正则性是这个研究项目的主题，我研究了Castelnuovo-Mumford正则性的上界的其他不变量的品种。Castelnuovo-Mumford正则性是使簇的理想层在Murnford意义下是m-正则的最大整数m，它决定了簇的定义方程的阶和簇的定义理想的合点。黎俊和我用所谓的卡斯泰尔诺沃界给出了一个关于正则性上界的猜想。我们还证明了极值情况只发生在有理正规涡卷上的因子的次数足够大的情况下。对于曲线的情况下，我已经得到了一个极端的界限，表明一般超平面部分的极值曲线包含在一个合理的正常曲线。经典的Caltelnuovo方法对于正特征情形的证明起着重要作用。此外，我已经分类的所有极端品种之间的约数的限制理性正常卷轴。这些事实应该能猜出推测。我想把它们推广到加权射影空间的相应结果。

My research has been devoted to the study of minimal free resolutions for the coordinate rings of projective varieties. In particular, the Castelnuovo-Mumford regularity is the main theme of this research project, and I have studied the upper bounds of the Castelnuovo-Mumford regularity in terms of the other invariants of give varieties. The Castelnuovo-Mumford regularity is the maximal integer m such that the ideal sheaf of the varietiy is m-regular in Murnford sense, and it regulates the degree of the defining equations of the variety and the syzygies of the defining ideal of the variety. Le Tuan Hoa and I had given a conjecture on an upper bound of the regularity by using so-called the Castelnuovo bound. Also we had conjectured the extremal case happens only in the case of a divisor on a rational normal scroll for its degree large enough. For the curve case, I have obtained an extremal bound by showing that the generic hyperplane section of an extremal curve is contained in a rational normal curve. The classical Caltelnuovo method plays an important role for the proof in positive characteristic case. Moreover, I have classified all the extremal varieties for the bound among the divisors on rational normal scrolls. These facts should guess the conjecture. I would like to extend them to the corresponding results for weighted projective space.