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The Picard group of the universal abelian variety and the Franchetta conjecture for abelian varieties. (English) Zbl 1430.14019
Over the moduli stack \(\mathcal{A}_{g,n}\), of principally polarized abelian varieties of dimension \(g\) with symplectic principal level-\(n\) structure, let \(\mathcal{X}_{g,n}\) be the universal abelian variety. The main result of the article under review pertains to the structure of the quotient sheaf \[ \operatorname{Pic}(\mathcal{X}_{g,n}) / \operatorname{Pic}(\mathcal{A}_{g,n}). \] It states that if \(g \geq 4\) and \(n \geq 1\), then \[ \operatorname{Pic}(\mathcal{X}_{g,n}) / \operatorname{Pic}(\mathcal{A}_{g,n})= \begin{cases} \left( \mathbb{Z} / n \mathbb{Z} \right)^{2g} \oplus \mathbb{Z}[\sqrt{\mathcal{L}_{\Lambda}}] & \text{ if }n\text{ is even} \\ \left( \mathbb{Z} / n \mathbb{Z} \right)^{2g} \oplus \mathbb{Z}[\mathcal{L}_{\Lambda} ] & \text{ if }n\text{ is odd.} \end{cases} \] Here, \(\sqrt{\mathcal{L}_{\Lambda}}\) is, up to torsion, a square-root of the rigidified canonical line bundle \(\mathcal{L}_{\Lambda}\) and \((\mathbb{Z} / n \mathbb{Z})^{2g}\) is the group of rigidified \(n\)th-roots of line bundles.
In a related direction, the authors study the quotient sheaf \[ \operatorname{Pic}(\mathcal{X}_g) / \operatorname{Pic}(\mathcal{A}_g) \] for \( \mathcal{X}_g \rightarrow \mathcal{A}_g\) the universal abelian variety over the moduli stack of principally polarized abelian varieties of dimension \(g\). In this context, they prove that it may be described as \[ \operatorname{Pic}(\mathcal{X}_g) / \operatorname{Pic}(\mathcal{A}_g) = \mathbb{Z}[\mathcal{L}_{\Lambda}] . \] The authors’ proof of their results builds on earlier work of A. Silverberg [Invent. Math. 81, 71–106 (1985; Zbl 0576.14020)] and D. Edidin and W. Graham [Invent. Math. 131, No. 3, 595–644 (1998; Zbl 0940.14003)].
At the same time, the present work is motived by earlier work of E. Arbarello and M. Cornalba [Topology 26, 153–171 (1987; Zbl 0625.14014)], M. Mestrano [Invent. Math. 87, 365–376 (1987; Zbl 0585.14011)] and A. Kouvidakis [J. Differ. Geom. 34, No. 3, 839–850 (1991; Zbl 0780.14004)].

MSC:
14C22 Picard groups
14K10 Algebraic moduli of abelian varieties, classification
14D20 Algebraic moduli problems, moduli of vector bundles
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