论文标题
$β$ - 密度的功能在集体组合的旋转复合品品种上
$β$-density function on the class group of projective toric varieties
论文作者
论文摘要
我们证明存在一个紧凑的支持，连续（有限多点除外）功能$ g_ {i，{\ bf m}}}：[0，\ infty）\ longrightArrow \ longrightArrow \ mathbb {r} $ for Allomial Prime prime $ r $ y of Height of Allomial Prime $ r $ a的$ r $ reptirance a $ r，co in $ r，r，r，r，r，\ bf m} \ bf mf m} \ bf m}折叠对$（x，d）$，使得$ \ int_ {0}^{\ infty} g_ {i，{\ bf m}}}（λ）dλ=β（i，{\ bf m}），$β（i，{\ bf m}）$ secons $ section $ s $ s $ s $ s $ s nikect of s $ iptiest $ iptiest y。最大理想$ {\ bf m} $，如Huneke-McDermott-Monsky \ Cite {HMM2004}所证明的。使用上面的结果，对于标准分级的正常仿型单子环，我们给出了类映射$τ_ {\ bf m}的完整描述：\ text {cl}（r）\ longrightArrow \ Mathbb {r {r} $在\ cite {hmm2004}中引入了第二个系数，以证明其第二个系数的存在。此外，我们显示函数$ g_ {i，{\ bf m}} $在segre产品上具有乘法性，其表达式涉及戒指和理想的希尔伯特多项式的前两个系数。 \关键字{Hilbert-kunz功能的系数\和投射旋转的旋转品种\和Hilbert-Kunz密度函数\和$β$ - 密度函数\和高度的单元理想。}
We prove the existence of a compactly supported, continuous (except at finitely many points) function $g_{I, {\bf m}}: [0, \infty)\longrightarrow \mathbb{R}$ for all monomial prime ideals $I$ of $R$ of height one where $(R, {\bf m})$ is the homogeneous coordinate ring associated to a projectively normal toric pair $(X, D)$, such that $$\int_{0}^{\infty}g_{I, {\bf m}}(λ)dλ=β(I, {\bf m}),$$ where $β(I, {\bf m})$ is the second coefficient of the Hilbert-Kunz function of $I$ with respect to the maximal ideal ${\bf m}$, as proved by Huneke-McDermott-Monsky \cite{HMM2004}. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map $τ_{\bf m}:\text{Cl}(R)\longrightarrow \mathbb{R}$ introduced in \cite{HMM2004} to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function $g_{I, {\bf m}}$ is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert polynomial of the rings and the ideals. \keywords{coefficients of Hilbert-Kunz function\and projective toric variety\and Hilbert-Kunz density function\and $β$-density function\and monomial prime ideal of height one.}
