The universal abelian variety over \(\mathcal{A}_5\) (Q2811186)

In the article under review, the authors study the universal abelian variety \(\mathcal{X}_5\) over \(\mathcal{A}_5\), the moduli stack of principally polarized abelian varieties of dimension \(5\). The authors' main result is that the universal abelian variety \(\mathcal{X}_5\) is unirational. In their proof of this result, the authors first establish a number of additional results which are of independent interest.NEWLINENEWLINEFor instance, the authors first study the universal Prym curve \(\widetilde{\mathcal{C}}\) over \(\overline{\mathcal{R}}_6\), the moduli space of stable Prym curves of genus \(6\), and its relation to \(\mathbf{G}\), the Hilbert scheme of lines in the fibres of the first projection \(\mathbb{P}^2 \times \mathbb{P}^2 \rightarrow \mathbb{P}^2\). In this setting, the authors define a rational map from \(\mathbf{G}^5\), the five fold product of \(\mathbf{G}\), to \(\widetilde{\mathcal{C}}^5\), the five fold product of \(\widetilde{\mathcal{C}}\). The authors then prove that this rational map \(\mathbf{G}^5 \dashrightarrow \widetilde{\mathcal{C}}^5\) is dominant from which they deduce that \(\widetilde{\mathcal{C}}^5\) is unirational.NEWLINENEWLINEAs one further consequence of this result, the authors use it to study the slope of the effective cone of \(\overline{\mathcal{A}}_6\), the first Voronoi compactification of \(\mathcal{A}_6\). In this direction, the authors prove that this slope is at least \(53/10\).