The Picard group of the compactified universal Jacobian
This paper describes the group of line bundles, or the Picard group, of the moduli space of line bundles on smooth curves as well as some closely related moduli spaces. The Picard group had been studied before, notably in [\textit{J. Ebert} and \textit{O. Randal-Williams}, Doc. Math., J. DMV 17, 417--450 (2012; Zbl 1273.14058); \textit{C. Fontanari}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 16, No. 1, 45--59 (2005; Zbl 1222.14055)], and [\textit{A. Kouvidakis}, J. Differ. Geom. 34, No. 3, 839--850 (1991; Zbl 0780.14004)], but this is the first paper to carefully treat stack-theoretic issues.NEWLINENEWLINERecall the moduli space of degree \(d\) line bundles on smooth curves of genus \(g\), denoted \(\mathcal{J}ac_{d,g}\) in the paper, exists as an algebraic stack. This stack has the property that every stabilizer group contains the multiplicative group \(\mathbb{G}_{m}\) (acting by scalar multiplication on line bundles), and a natural rigidification construction removes the \(\mathbb{G}_{m}\)s to produce a new stack \(\mathcal{J}_{d,g}\). The stabilizer groups of the new stack \(\mathcal{J}_{d,g}\) are finite groups, in fact \(\mathcal{J}_{d,g}\) is a Deligne-Mumford stack, but the stabilizers can be nontrivial, so \(\mathcal{J}_{d,g}\) is not an algebraic variety. The nontrivial stabilizer groups can, however, be removed to produce a third object, the coarse moduli space of \(\mathcal{J}_{d,g}\) which is a quasi-projective variety. In the literature, all of these spaces are often called the universal Jacobian, although in the present paper the term is reserved for \(\mathcal{J}ac_{d,g}\).NEWLINENEWLINEThe universal Jacobian is studied alongside certain moduli spaces parameterizing line bundles on (possible unstable) nodal curves that satisfy the balancedness condition (a numerical condition on the multidegree). Similar to the previous situation, one considers the moduli stack \(\overline{\mathcal{J}ac}_{d,g}\) of these objects as well as a Deligne-Mumford stack \(\overline{\mathcal{J}}_{d,g}\) obtained by removing \(\mathbb{G}_{m}\)s from stabilizers and a projective variety \(\overline{J}_{d,g}\) obtained by removing all stabilizers. All of these spaces are often called the compactified universal Jacobian as \(\overline{J}_{d,g}\) is proper.NEWLINENEWLINEThe main results of this paper compute the Picard groups of these stacks when \(g \geq 3\). The results are easiest to state for \(\mathcal{J}ac_{d,g}\) and \(\overline{\mathcal{J}ac}_{d,g}\). The universal family of curves over \(\mathcal{J}ac_{d,g}\) admits two natural line bundles: the universal line bundle \(\mathcal{L}\) and the relative dualizing sheaf \(\omega\). Theorem A states that the Picard group of \(\mathcal{J}ac_{d,g}\) is freely generated by the determinants of cohomology of \(\mathcal{L}\), \(\omega\), and \(\mathcal{L} \otimes \omega\). Furthermore, the Picard group of \(\overline{\mathcal{J}ac}_{d,g}\) is freely generated by these line bundles together with the line bundles associated to the irreducible components of the boundary (i.e. the complement of \(\mathcal{J}ac_{d,g}\) in \(\overline{\mathcal{J}ac}_{d,g}\)).NEWLINENEWLINEThe Picard groups of \(\mathcal{J}_{d,g}\) and \(\overline{\mathcal{J}}_{d,g}\) are described in Theorem B. This result is more complicated to state because the universal family of curves over \(\mathcal{J}_{d,g}\) does not admit a universal family of line bundles. The result states that the Picard group of \(\mathcal{J}_{d,g}\) is freely generated by the determinant of cohomology of \(\omega\) and a line bundle that is more complicated to describe but is similar to a certain explicit linear combination of the determinants of cohomology of \(\mathcal{L}\) and \(\mathcal{L} \otimes \omega\). These line bundles together with the line bundles associated to the irreducible components of the boundary freely generate the Picard group of \(\overline{\mathcal{J}}_{d,g}\).NEWLINENEWLINEThe analogous results for \(J_{d,g}\) and \(\overline{J}_{d,g}\) were known by [Zbl 1222.14055], and the authors also relate the descriptions in that work to their Theorems A and B in Theorem C.NEWLINENEWLINEThe main theorems are proven by using Kouvidakis's work [Zbl 0780.14004] to compute the Picard group of \(\mathcal{J}_{d,g}\) and then relating the other Picard groups of interest to \(\text{Pic}(\mathcal{J}_{d,g})\). Results similar to the results about \(\mathcal{J}ac_{d,g}\) and \(\mathcal{J}_{d,g}\) were proven in [Zbl 1273.14058], which appeared shortly after a preliminary of the present paper was made publically available. The authors of that paper compute the Picard groups of topological stacks that are expected to be topological models of \(\mathcal{J}ac_{d,g}\) and \(\mathcal{J}_{d,g}\). The proofs there are very different from the proofs in the present paper. In [Zbl 1273.14058] the main results are proven using algebraic topology, especially ideas from homotopy theory, while the present paper uses algebraic geometry. The relation between the results of the two papers is carefully described in Section 1.1 of the present paper and Section 4.5 of [Zbl 1273.14058].
scientific article; zbMATH DE number 6358800