Free resolutions of the coordinate rings of projective varieties

我的研究一直致力于射影变量坐标环的最小自由分辨率的研究。特别地，Castelnuovo-Mumford正则性是本研究项目的主题，我已经根据给定变量的其他不变量研究了Castelnuovo-Mumford正则性的上界。Castelnuovo-Mumford正则性是使品种理想束在Murnford意义上具有m正则性的最大整数m，它调节了品种定义方程的程度和品种定义理想的协同性。我和Le Tuan Hoa用所谓的Castelnuovo界给出了正则性上界的一个猜想。我们还推测，极端情况只发生在有理法向卷轴上的除数足够大的情况下。对于曲线的情况，我通过证明极值曲线的一般超平面截面包含在有理正态曲线中，得到了一个极值界。经典Caltelnuovo方法在正特征情况下的证明中起着重要的作用。此外，我还对有理法向卷轴上的除数之间的界的所有极值变体进行了分类。这些事实应该能证明这个猜想。我想把它们推广到加权投影空间的相应结果。

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.