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arXiv 1808.04093math.ACmath.AG
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Hilbert-Kunz密度函数与$F$阈值

Hilbert-Kunz density functions and $F$-thresholds

Vijaylaxmi Trivedi, Kei-Ichi Watanabe

备注：
23 pages, This paper is the first part of arXiv:1808.04093, which is now divided into two parts. The part containing exclusively the two dimensional case has been removed and will be posted as another paper
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摘要：

我们在前面已经证明了，对于标准分次环$R$和特征$p>0$中的分次理想$I$，当有$\ell(R/I) <\infty$时，存在一个紧支集连续函数$f_{R, I}$，其黎曼积分是HK重数$e_{HK}(R, I)$。我们进一步探讨了其他一些不变量，即图的形状$f_{R, {\bf m}}$（其中${\bf m}$是$R$的分次极大理想）和$f_{R, I}$的最大支持度（记为$α(R,I)$）。在$R$是$d\geq 2$维的整环的情况下，我们证明了$(R, {\bf m})$是正则环当且仅当对所有的$x$，$f_{R, {\bf m}}$有对称性$f_{R, {\bf m}}(x) = f_{R, {\bf m}}(d-x)$.如果$R$是强$F$ -正规的打孔频谱，然后我们证明了$F$ -阈值$c^I({\bf m})$与$α(R,I)$一致。因此，如果$R$是一个二维区域，并且$I$是由相同次数的齐次元素生成的，则有（1）根据合合点丛的最小强Harder-Narasimahan斜率的$F$阈值$c^I({\bf m})$的公式和（2）在特征$0$中定义良好的$F$阈值$c^I({\bf m})$的概念。这个特征很容易计算$c^{I(n)}({\bf m})$，对于所有不可约的平面三项式$k[x,y,z]/(h)$的集合，其中${\bf m} = (x,y,z)$和$I(n) = (x^n, y^n, z^n)$。

英文摘要：

We had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $\ell(R/I) <\infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK multiplicity $e_{HK}(R, I)$. We explore further some other invariants, namely the shape of the graph of $f_{R, {\bf m}}$ (where ${\bf m}$ is the graded maximal ideal of $R$) and the maximum support (denoted as $α(R,I)$) of $f_{R, I}$. In case $R$ is a domain of dimension $d\geq 2$, we prove that $(R, {\bf m})$ is a regular ring if and only if $f_{R, {\bf m}}$ has a symmetry $f_{R, {\bf m}}(x) = f_{R, {\bf m}}(d-x)$, for all $x$. If $R$ is strongly $F$-regular on the punctured spectrum then we prove that the $F$-threshold $c^I({\bf m})$ coincides with $α(R,I)$. As a consequence, if $R$ is a two dimensional domain and $I$ is generated by homogeneous elements of the same degree, thene have (1) a formula for the $F$-threshold $c^I({\bf m})$ in terms of the minimum strong Harder-Narasimahan slope of the syzygy bundle and (2) a well defined notion of the $F$-threshold $c^I({\bf m})$ in characteristic $0$. This characterisation readily computes $c^{I(n)}({\bf m})$, for the set of all irreducible plane trinomials $k[x,y,z]/(h)$, where ${\bf m} = (x,y,z)$ and $I(n) = (x^n, y^n, z^n)$.

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。

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