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تسجيل مستخدم جديدHilbert-Kunz density functions and $F$-thresholds
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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 $alpha(R,I)$) of $f_{R, I}$. In case $R$ is a domain of dimension $dgeq 2$, 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 $alpha(R,I)$. 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 $I(n) = (x^n, y^n, z^n)$.
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For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $ell(R/I) < infty$, we introduce a new invariant $HKd(M, I)$ called the {em Hilbert-Kunz density funct
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ivalent, for any fixed $mgeq d-1$, to the existence of $lim_{pto infty}ell(R_p/I_p^{[p^m]})/p^{md}$. This we get as a consequence of Theorem 1.1: As $prightarrow infty $, the convergence of the HK density function $f{(R_p, I_p)}$ is equivalent to the convergence of the truncated HK density functions $f_m(R_p, I_p)$ (in $L^{infty}$ norm) of the {it mod $p$ reductions} $(R_p, I_p)$, for any fixed $mgeq d-1$. In particular, to define the HK density function $f^{infty}(R, I)$ in characteristic 0, it is enough to prove the existence of $lim_{pto infty} f_m(R_p, I_p)$, for any fixed $mgeq d-1$. This allows us to prove the existence of $e_{HK}^{infty}(R, I)$ in many new cases, {em e.g.}, when $mbox{Proj~R}$ is a Segre product of curves, for example.
We prove the existence of HK density function for a pair $(R, I)$, where $R$ is a ${mathbb N}$-graded domain of finite type over a perfect field and $Isubset R$ is a graded ideal of finite colength. This generalizes our earlier result where one prove
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Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptio
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