Let \(X\) by a smooth projective variety, \(\mathfrak{a}\) an ideal sheaf on \(X\) and \(\pi: Y\to X\) a log resolution of the pair \((X,\mathfrak{a})\). Let \(F\) be the Cartier divisor such that \(\mathfrak{a}\cdot\mathcal{O}_Y=\mathcal{O}_Y(-F)\) and \(K_{\pi}=K_Y-\pi^*K_X\). For a non-negative number \(c\), one can associate to \(\mathfrak{a}\) a multiplier ideal sheaf \(\mathcal{J}(X, \mathfrak{a}^c)=\pi_*\mathcal{O}_Y(K_\pi-\lfloor cF\rfloor)\) which measures the complexity of the singularity of \(\mathfrak{a}\). When the values of \(c\) increases, the multiplier ideal may change. The values of \(c\) such that the multiplier ideal changes are known as the jumping numbers of the pair \((X, \mathfrak{a})\).
In the paper under review, the authors generalize the algorithm for computing the jumping numbers on rational surfaces in [M. Alberich-Carramiñana et al., Michigan Math. J. 65, No. 2, 287–320 (2016; Zbl 1357.14025)] to the higher dimensional varieties. The algorithm starts with computing supercandidates and then check whether they are jumping number.
To compute supercandiates, which forms a set containing the set of jumping numbers (Theorem 4.4), the authors first generalize \(\pi\)-antinef divisors on surfaces to \(\pi\)-antieffective divisors to higher dimension and show that every divisor \(D\) has a unique \(\pi\)-antieffective closure (Theorem 3.4). Starting with the first supercandiate, the log canonical threshold which is the smallest jumping number, the set of supercandidates is constructed inductively by adding the number \(\min\limits_{i}\left\{\frac{k_1+1+e_i^{\lambda}}{e_i}\right\}\) if \(D_\lambda = \sum\limits_i e_i^{\lambda}E_i\) is the \(\pi\)-antieffective closure of \(\lfloor cF\rfloor-K_\pi\). Associate to a supercandidate \(\lambda\), the authors define the minimal jumping divisor \(G_\lambda\) to be the reduced divisor supported on the components \(E_i\) where \(\min\limits_{i}\left\{\frac{k_1+1+e_i^{\lambda}}{e_i}\right\}=\frac{k_1+1+e_i^{\lambda}}{e_i}\).
A necessary condition that a supercandidate \(\lambda\) is a jumping number is that the minimal jumping divisor \(G_\lambda\) contributes \(\lambda\) (Proposition 4.7) that is \(\mathcal{J}(X,\mathfrak{a}^\lambda)\subsetneq \pi_*\mathcal{O}_Y(K_\pi-\lfloor \lambda F\rfloor + G_\lambda)\). A sufficient condition is that \(G_\lambda\) has a connected component which is irreducible. Based on this criterion and the idea of looking for contributing divisors, the authors propose algorithms to determine whether a supercandidate is a jumping number in different situations. However, due to the unknown correspondence between \(\pi\)-antieffective divisors and integrally closed ideals in higher dimensions, it is not known whether all supercandidates are actual jumping numbers.
The authors also provide examples on how to implement the algorithm.