Waldschmidt constants for Stanley-Reisner ideals of a class of simplicial complexes (Q2816901)

In the present paper, the authors study the so-called symbolic powers of the Stanley-Reisner ideal \(I_{B_{n}}\) of a bipyramid \(B_{n}\) over an \(n\)-gon \(Q_{n}\). Let us recall some basic definitions. Let \(k\) be a field and suppose that \(I \leq k[x_{0},\dots,x_{n}]\) is a non-trivial homogeneous ideal. Then the \(m\)-th symbolic power of \(I\) can be defined as NEWLINE\[NEWLINEI^{(m)} = R \cap \bigcap_{P \in \mathrm{Ass}(I)}I^{m}R_{p},NEWLINE\]NEWLINE where the intersection is taken in the field of fractions and \(\mathrm{Ass}(I)\) denotes the set of associated primes of \(I\). The containment problem concerns with the study of pairs \((m,r)\) such that \(I^{(m)} \subset I^{r}\), where \(I^{r}\) denotes the \(r\)-th ordinary power of \(I\). One of the most surprising result is due to Ein-Lazarsfeld-Smith [\textit{L. Ein} et al., Invent. Math. 144, No. 2, 241--252 (2001; Zbl 1076.13501)] and Hochster-Huneke [\textit{M. Hochster} and \textit{C. Huneke}, Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)] stating that \(I^{(er)} \subset I^{r}\), where \(e\) is the codimension of \(I\). A natural approach to the containment problem is to use asymptotic invariants. Among them there is the so-called Waldschmidt constant, denoted by \(\widehat{\alpha}(I)\), defined as NEWLINE\[NEWLINE\widehat{\alpha}(I) = \lim_{m \rightarrow \infty} \frac{\alpha(I^{(m)})}{m},NEWLINE\]NEWLINE where \(\alpha(I) = \min \{t : I_{t}^{(m)} \neq 0\}\). It is worth pointing out that in general it is really difficult to compute Waldschmidt contants.NEWLINENEWLINEIn the present paper the authors focus on a special class of the square-free monomial ideals, i.e., the Stanley-Reisner ideals \(I_{B_{n}}\) of bypiramids \(B_{n}\) over \(n\)-gons. The key asset of this class is that one can use a purely combinatorial approach to compute the \textit{initial degree} \(\alpha({I^{(m)}_{B_{n}}})\). The main result of the paper can be formulated as follows.NEWLINENEWLINETheorem. Let \(I_{B_{n}}\) be the Stanley-Reisner ideal of a bipyramid over an \(n\)-gon, then for every \(n \in \mathbb{Z}_{>0}\) one has NEWLINE\[NEWLINE\widehat{\alpha}(I_{B_{n}}) = \frac{n}{n-2}.NEWLINE\]NEWLINENEWLINENEWLINEThe paper is nicely written and self-contained.