The slope of v-function and Waldschmidt constant
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916226238775296 |
|---|---|
| author | Kumar, Manohar Nanduri, Ramakrishna Saha, Kamalesh |
| author_facet | Kumar, Manohar Nanduri, Ramakrishna Saha, Kamalesh |
| contents | In this paper, we study the asymptotic behaviour of the v-number of a Noetherian graded filtration $\mathcal{I}= \{I_{[k]}\}_{k\geq 0}$ of a Noetherian $\mathbb{N}$-graded domain $R$. Recently, it is shown that $\mathrm{v}(I_{[k]})$ is periodically linear in $k$ for $k \gg 0$. We show that all these linear functions have the same slope, i.e. $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I_{[k]})}{k}$ exists, which is equal to $\displaystyle \lim_{k \rightarrow \infty}\frac{α(I_{[k]})}{k}$, where $α(I)$ denotes the minimum degree of a non-zero element in $I$. In particular, for any Noetherian symbolic filtration $\mathcal{I}= \{I^{(k)}\}_{k\geq 0}$ of $R$, it follows that $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I^{(k)})}{k}=\hatα(I)$, the Waldschmidt constant of $I$. Next, for a non-equigenerated square-free monomial ideal $I$, we prove that $\mathrm{v}(I^{(k)}) \leq \mathrm{reg}(R/I^{(k)})$ for $k\gg 0$. Also, for an ideal $I$ having the symbolic strong persistence property, we give a linear upper bound on $\mathrm{v}(I^{(k)})$. As an application, we derive some criteria for a square-free monomial ideal $I$ to satisfy $\mathrm{v}(I^{(k)})\leq \mathrm{reg}(R/I^{(k)})$ for all $k\geq 1$, and provide several examples in support. In addition, for any simple graph $G$, we establish that $\mathrm{v}(J(G)^{(k)}) \leq \mathrm{reg}(R/J(G)^{(k)})$ for all $k \geq 1$, and $\mathrm{v}(J(G)^{(k)}) = \mathrm{reg}(R/J(G)^{(k)})=α(J(G)^{(k)})-1$ for all $k\geq 1$ if and only if $G$ is a Cohen-Macaulay very-well covered graph, where $J(G)$ is the cover ideal of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The slope of v-function and Waldschmidt constant Kumar, Manohar Nanduri, Ramakrishna Saha, Kamalesh Commutative Algebra Combinatorics 13F20, 13A02, 13F55, 05E40 In this paper, we study the asymptotic behaviour of the v-number of a Noetherian graded filtration $\mathcal{I}= \{I_{[k]}\}_{k\geq 0}$ of a Noetherian $\mathbb{N}$-graded domain $R$. Recently, it is shown that $\mathrm{v}(I_{[k]})$ is periodically linear in $k$ for $k \gg 0$. We show that all these linear functions have the same slope, i.e. $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I_{[k]})}{k}$ exists, which is equal to $\displaystyle \lim_{k \rightarrow \infty}\frac{α(I_{[k]})}{k}$, where $α(I)$ denotes the minimum degree of a non-zero element in $I$. In particular, for any Noetherian symbolic filtration $\mathcal{I}= \{I^{(k)}\}_{k\geq 0}$ of $R$, it follows that $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I^{(k)})}{k}=\hatα(I)$, the Waldschmidt constant of $I$. Next, for a non-equigenerated square-free monomial ideal $I$, we prove that $\mathrm{v}(I^{(k)}) \leq \mathrm{reg}(R/I^{(k)})$ for $k\gg 0$. Also, for an ideal $I$ having the symbolic strong persistence property, we give a linear upper bound on $\mathrm{v}(I^{(k)})$. As an application, we derive some criteria for a square-free monomial ideal $I$ to satisfy $\mathrm{v}(I^{(k)})\leq \mathrm{reg}(R/I^{(k)})$ for all $k\geq 1$, and provide several examples in support. In addition, for any simple graph $G$, we establish that $\mathrm{v}(J(G)^{(k)}) \leq \mathrm{reg}(R/J(G)^{(k)})$ for all $k \geq 1$, and $\mathrm{v}(J(G)^{(k)}) = \mathrm{reg}(R/J(G)^{(k)})=α(J(G)^{(k)})-1$ for all $k\geq 1$ if and only if $G$ is a Cohen-Macaulay very-well covered graph, where $J(G)$ is the cover ideal of $G$. |
| title | The slope of v-function and Waldschmidt constant |
| topic | Commutative Algebra Combinatorics 13F20, 13A02, 13F55, 05E40 |
| url | https://arxiv.org/abs/2404.00493 |