The paper under review obtains a lower bound for the minimal slope of effective divisors on \(\mathcal{A}_5\), the moduli space of principally polarized abelian \(5\)-folds, by computing the pull back of divisor classes under the Prym map. It is a very good example of how to study the slopes of effective cones of moduli spaces.
The correspondence \[ \pi:\overline{\mathcal{M}}_g\leftarrow\overline{\mathcal{R}}_g\dashrightarrow\overline{\mathcal{A}}_{g-1}:p \] given by the moduli of Prym curves plays an essential role in the paper. Here \(\overline{\mathcal{A}}_{g-1}\) is the perfect cone toroidal compactification whose Picard group is generated by the Hodge class \(L\) and the boundary class \(D\).
The main result (Theorem 6) of the paper is the computation of the pull back \[ p^*:\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{A}}_{g-1})\to \mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{R}}_g) \] for \(g\geqslant 6\). This is carried out in Section 3. The key is a beautiful use of the Schottky-Jung proportionality over \(\overline{\mathcal{R}}_g\) to compute the pull back of the theta null divisor. The formulas for \(\pi_*p^*\) follow easily from the main result.
As an application, one can make use of the known results about the slopes of the effective divisors on \(\overline{\mathcal{M}}_g\). For \(g=6\), \(p\) is dominant and generically finite. Therefore the lower bound of the slopes for \(\overline{\mathcal{M}}_6\) gives a lower bound of the slopes for \(\overline{\mathcal{A}}_5\). That is \(\frac{a}{b} \geqslant 7+\frac{4198}{6269}\), if \(a,b>0\) and \(aL-bD\) is an effective divisor on \(\overline{\mathcal{A}}_5\). The appendix discusses why the slope for \(\overline{\mathcal{A}}_{g}\) should be regarded as the slope for \(\mathcal{A}_{g}\), i.e., the result here is independent of the choice of the toroidal compactification.