arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 2007.12394math.AGmath.AC
收藏

$F$ -射影曲线的阈值$c^I({\bf m})$

$F$-thresholds $c^I({\bf m})$ for projective curves

Vijaylaxmi Trivedi

备注：
21 pages, This paper is the second part of the paper arXiv:1808.04093 v1, which is now divided into two parts. The first part is now posted as arXiv:1808.04093 v2
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

我们证明了如果$R$是一个二维标准分次环（具有分次极大理想${\bf m}$）的特征$p>0$和$I\subset R$是一个分次理想与$\ell(R/I) <\infty$然后$F$阈值$c^I({\bf m})$可以表示在一个强HN（Harder-Narasimahan）斜率的正则syzygy丛$\mbox{Proj}~R$。因此$c^I({\bf m})$是有理数。这给了我们一个很好定义的概念，$F$ -在特征$0$中的阈值$c^I({\bf m})$，根据$\mbox{Proj}~R$上的合冲束的HN斜率。这推广了我们先前的结果（在[TrW]），其中我们已经表明，如果$I$具有相同程度的齐次生成元，则$F$ -阈值$c^I({\bf m})$分别用$\mbox{Proj}~R$上的正则朔望丛的最小强HN斜率（在char $p$中）和最小HN斜率（在char $0$中）表示。在这里，我们还证明，对于一个给定的对$(R, I)$在一个领域的特征$0$，如果$({\bf m}_p, I_p)$是一个减少模$p$的$({\bf m}, I)$然后$c^{I_p}({\bf m}_p) \neq c^I_{\infty}({\bf m})$意味着$c^{I_p}({\bf m}_p)$有$p$的分母，几乎所有的$p$。

英文摘要：

We show that if $R$ is a two dimensional standard graded ring (with the graded maximal ideal ${\bf m}$) of characteristic $p>0$ and $I\subset R$ is a graded ideal with $\ell(R/I) <\infty$ then the $F$-threshold $c^I({\bf m})$ can be expressed in terms of a strong HN (Harder-Narasimahan) slope of the canonical syzygy bundle on $\mbox{Proj}~R$. Thus $c^I({\bf m})$ is a rational number. This gives us a well defined notion, of the $F$-threshold $c^I({\bf m})$ in characteristic $0$, in terms of a HN slope of the syzygy bundle on $\mbox{Proj}~R$. This generalizes our earlier result (in [TrW]) where we have shown that if $I$ has homogeneous generators of the same degree, then the $F$-threshold $c^I({\bf m})$ is expressed in terms of the minimal strong HN slope (in char $p$) and in terms of the minimal HN slope (in char $0$), respectively, of the canonical syzygy bundle on $\mbox{Proj}~R$. Here we also prove that, for a given pair $(R, I)$ over a field of characteristic $0$, if $({\bf m}_p, I_p)$ is a reduction mod $p$ of $({\bf m}, I)$ then $c^{I_p}({\bf m}_p) \neq c^I_{\infty}({\bf m})$ implies $c^{I_p}({\bf m}_p)$ has $p$ in the denominator, for almost all $p$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑