In the paper under review, the author studies the so-called containment problem from a viewpoint of the Harbourne-Huneke bound. Let us recall that a groundbreaking result of L. Ein et al. [Invent. Math. 144, No. 2, 241–252 (2001; Zbl 1076.13501)] in characteristic zero and M. Hochster and C. Huneke [Invent. Math. 147, No. 2, 349–369 (2002; Zbl 1061.13005)] in positive characteristic implies that the symbolic power \[ I^{(Nr)} \subseteq I^{r} \] for all graded non-trivial ideals \(I \subset \mathbb{F}[\mathbb{P}^{N}] = \mathbb{F}[x_{0}, \dots, x_{N}]\) and all integers \(r >0\). In particular, for \(\mathbb{F}[\mathbb{P}^{2}]\) one always has that \(I^{(4)} \subset I^{2}\). In the meantime, Huneke asked whether one has \(I^{(3)} \subset I^{2}\). Following this idea, Harbourne generalized Huneke’s statement to \[ (\star) : \quad I^{(N(r-1)+1)} \subseteq I^{r} \] for any graded non-trivial ideal \(I \subset \mathbb{F}[\mathbb{P}^{N}]\), all \(r\geq 1\), and all \(N\geq 2\). However, it turned out that Harbourne-Huneke’s prediction is not quite right, the first counterexample to the containment \(I^{(3)} \subset I^{2}\) was found by M. Dumnicki et al. in [J. Algebra 393, 24–29 (2013; Zbl 1297.14008)]. In the paper under review, the author revisits the problem around the Harbourne-Huneke bound and it is shown that the containment \((\star)\) holds for certain classes of ideals in certain non-regular rings. More preciesly, the author shows the following results (for needed definitions, please consult Section 2 and 3 therein).
{Theorem A}. Let \(R_{1}, \dots, R_{n}\) be normal affine semigroup rings over a field \(\mathbb{F}\). For each \(1 \leq i \leq n\) suppose there is an integer \(D_{i} > 0\) such that \(P^{(D_{i}(r-1)+1)} \subseteq P^{r}\) for all \(r>0\) and all monomial primes \(P \subseteq R_{i}\). Set \(D = \max \{D_{1}, \dots,D_{n}\}\). Then \(Q^{(D(r-1)+1)} \subseteq Q^{r}\) for all \(r>0\) and any monomial prime \(Q\) in the normal affine semigroup ring \(R = R_{1}\otimes_{\mathbb{F}} \dots \otimes_{\mathbb{F}} R_{n}\).
{Theorem B}. Let \(S = \mathbb{F}[x_{1}, \dots, x_{n}]\) for \(n\geq 1\) be a polynomial ring over an arbitrary field \(\mathbb{F}\) and consider the module-finite extensions of normal toric rings \(V_{D} \subseteq S \subset H_{D}\), where
i) \(V_{D} \subseteq S\) is the \(D\)-th Veronese subring with its standard \(\mathbb{N}\)-grading, and
ii) \(H_{D} = \mathbb{F}[z,x_{1}, \dots, x_{n}] / (z^{D}-x_{1} - \dots - x_{n})\) is a hypersurface ring.
Then \(P^{(D(r-1)+1)} \subseteq P^{r}\) for all \(r>0\), where \(P\) is a monomial ideal in any of the three rings.