Special line arrangements in \(\mathbb {P}^2\) and Waldschmidt constant of their singular points
In the paper under review, the authors study the so-called Waldschmidt constants of homogeneous ideals of points in the projective plane associated with the singular loci of line arrangements. Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and consider a finite set of mutually distinct points $\mathcal{P} = \{P_{1}, \dots, P_{s}\} \subset \mathbb{P}^{2}_{\mathbb{K}}$. We define the associated ideal \[ I(\mathcal{P}) = \bigcap_{i=1}^{s}I(P_{i}). \] The $m$-th symbolic power of $I = I(\mathcal{P})$ is defined as \[ I^{(m)} = \bigcap_{i=1}^{s}I(P_{i})^{m}, \] which geometrically can be viewed as the set of all homogeneous polynomials that vanish to order at least $m$ at each of the points $P_{i}$. Moreover, we can consider the Waldschmidt constant of any non-trivial homogeneous ideal $I$ of $\mathbb{K}[\mathbb{P}^{2}]$ as \[ \widehat{\alpha}(I) = \text{lim}_{m \rightarrow \infty} \frac{\alpha(I^{(m)})}{m}, \] where $\alpha(\cdot)$ denotes the initial degree of $I$. The main result of the note is devoted to special point configurations $\mathcal{P}$ which are the singular loci of certain line arrangements. Let $\mathcal{L} \subset \mathbb{P}^{2}_{\mathbb{K}}$ be an arrangement of $d$ lines and denote by $\text{Sing}(\mathcal{L})$ the singular locus of $\mathcal{L}$. Assume that $r, d$ are two integers such that $3 \leq d$, $0 \leq 2r \leq d$, and $\mathcal{L} = \{L_{1}, \dots, L_{d}\}$ is an arrangement of $d$ lines such that its elements satisfy the following conditions: \begin{itemize} \item[i)] the lines $L_{1},\dots, L_{d-r}$ pass through a fixed point; \item[ii)] the lines $L_{i}$, where $1 \leq i \leq d-r$, and the lines $L_{j}$, where $d-r + 1 \leq j \leq d$, meet in exactly $(d-r)r$ points. \end{itemize} Main Theorem. Let $\mathcal{L}$ be an arrangement of lines in $\mathbb{P}^{2}_{\mathbb{K}}$ satisfying the above two conditions. Consider the radical ideal $I_{\text{Sing}(\mathcal{L})}$ associated with the singular locus of $\mathcal{L}$. Then \[ \widehat{\alpha}(I_{\text{Sing}(\mathcal{L})}) = r+1 - \frac{r}{d-r}. \]
scientific article; zbMATH DE number 7040721
scientific article