论文标题
$ f $ - 阈值$ c^i（{\ bf m}）$用于投影曲线
$F$-thresholds $c^I({\bf m})$ for projective curves
论文作者
论文摘要
我们表明，如果$ r $是特征的特征$ p> 0 $和$ i \ i \ subset r $的二维标准级别（具有最大理想$ {\ bf m} $），则是$ \ ell（r/i）<\ infty $ the $ f $ f $ f $ f $ c^i（r-el f $ c^i（r/el）的$ c^i（r/if）$ c^i（r/el）的标准$ c^i（r/I） （较硬的narasimahan）在$ \ mbox {proj} 〜r $上的典型syzygy捆绑包的斜率。因此，$ c^i（{\ bf m}）$是一个合理的数字。这为我们提供了一个明确定义的概念，即特征性$ 0 $的$ f $ threshold $ c^i（{\ bf m}）$，就$ \ mbox {proj} 〜r $的syzygy Bundle的hn斜率而言。这概括了我们的较早结果（在[trw]中），我们已经表明，如果$ i $具有同一度的同质发电机，那么$ f $ threshold $ c^i（{\ bf m}）$表示，根据最小的强力Hn Slope（在Char $ p $中）以及对最小的Hn Slope（在Char $ p $中的最小值）（在Bunnle car char $ char phope中）表示，在Bunnle car $ 0. 0中， $ \ mbox {proj} 〜r $。在这里，我们还证明，对于给定的$（r，i）$，在特征$ 0 $的字段上，如果$（{\ bf m} _p，i_p）$是一个还原的mod mod mod mod mod $ p $ of $（{\ bf m}，i），然后$ c^{i_p}（i_p}（\ bf m} _p） c^i _ {\ infty}（{\ bf m}）$暗示$ c^{i_p}（{\ bf m} _p）$在分母中具有$ p $，几乎所有$ p $。
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 $p$.
