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Moduli of roots of line bundles on curves. (English) Zbl 1140.14022
The paper under review focuses on the problem of compactifying the moduli space of roots of line bundles on curves. More precisely, let \(f:C\to B\) be a family of nodal curves with nonsingular fibers over an open subset \(U\subset B\). Let \(N\) be a line bundle on \(C\) whose degree is multiple of \(r\). Then, the family of line bundles on \(U\), \(L\), such that \(L^{\otimes r}=N\) is an étale covering of \(U\). The aim of the paper is to compactify over \(B\) such a covering. Previous works in the same direction have been carried out by A. Beauville [Invent. Math. 41, 149-196 (1977; Zbl 0333.14013)], M. Cornalba [in: Proc. first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 560–589 (1989; Zbl 0800.14011)], T. J. Jarvis [Compos. Math. 110, No. 3, 291–333 (1998; Zbl 0912.14010) and Int. J. Math. 11, No. 5, 637–663 (2000; Zbl 1094.14504)], D. Abramovich and T. J. Jarvis [Proc. Am. Math. Soc. 131, No. 3, 685–699 (2002; Zbl 1037.14008)] and A. Chiodo [“Higher Spin curves and Witten’s top Chern class”, Ph.D. thesis, University of Cambridge, (2003)]. Roughly speaking, in these constructions the boundary points corresponds to certain rank \(1\) torsion free sheaves on nodal curves or nodal curves with a stack structure at some nodes.
The first part of the paper, which follows the Cornalba’s ideas (cited above), introduces the notion of “limit \(r\)-th roots” and offers a general solution to the problem in terms of line bundles on nodal curves by constructing a coarse moduli space. In the “higher spin case”, that is, roots of \(\omega_f^{\otimes l}\), the resulting space is isomorphic to that given by Jarvis (loc. cit.). A second important result is that, for \(r\geq 3\), the previously constructed moduli spaces as well as those constructed by Abramovich-Jarvis, Chiodo and Jarvis (loc. cit.) is not finite over the moduli space of stable curves and do not embed in the universal Picard scheme (for the \(r=2\) case [see C. Fontanari, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16, No. 1, 45–59 (2005; Zbl 1222.14055)].

MSC:
14H10 Families, moduli of curves (algebraic)
14H60 Vector bundles on curves and their moduli
14K30 Picard schemes, higher Jacobians
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