The stability space of compactified universal Jacobians
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- by Jesse Leo Kass and Nicola Pagani PDF
- Trans. Amer. Math. Soc. 372 (2019), 4851-4887 Request permission
Abstract:
In this paper we describe compactified universal Jacobians, i.e., compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank $1$ torsion-free sheaves on stable curves, using an approach due to Oda–Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne–Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo) and to resolve the indeterminacy of the Abel–Jacobi sections (addressing a problem raised by Grushevsky–Zakharov).References
- Enrico Arbarello and Maurizio Cornalba, The Picard groups of the moduli spaces of curves, Topology 26 (1987), no. 2, 153–171. MR 895568, DOI 10.1016/0040-9383(87)90056-5
- Enrico Arbarello and Maurizio Cornalba, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 97–127 (1999). MR 1733327
- Allen B. Altman and Steven L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112. MR 555258, DOI 10.1016/0001-8708(80)90043-2
- Valery Alexeev, Compactified Jacobians and Torelli map, Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, 1241–1265. MR 2105707
- Gilberto Bini, Claudio Fontanari, and Filippo Viviani, On the birational geometry of the universal Picard variety, Int. Math. Res. Not. IMRN 4 (2012), 740–780. MR 2889156, DOI 10.1093/imrn/rnr045
- Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR 1217487, DOI 10.1007/978-3-662-02788-2
- Lucia Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994), no. 3, 589–660. MR 1254134, DOI 10.1090/S0894-0347-1994-1254134-8
- Lucia Caporaso, Néron models and compactified Picard schemes over the moduli stack of stable curves, Amer. J. Math. 130 (2008), no. 1, 1–47. MR 2382140, DOI 10.1353/ajm.2008.0000
- Renzo Cavalieri, Paul Johnson, and Hannah Markwig, Wall crossings for double Hurwitz numbers, Adv. Math. 228 (2011), no. 4, 1894–1937. MR 2836109, DOI 10.1016/j.aim.2011.06.021
- Sebastian Casalaina-Martin, Jesse Leo Kass, and Filippo Viviani, The local structure of compactified Jacobians, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 510–542. MR 3335286, DOI 10.1112/plms/pdu063
- Bashar Dudin, Compactified universal Jacobian and the double ramification cycle, Int. Math. Res. Not. IMRN 8 (2018), 2416–2446. MR 3801488, DOI 10.1093/imrn/rnw313
- David Eisenbud and Joe Harris, The Kodaira dimension of the moduli space of curves of genus $\geq 23$, Invent. Math. 90 (1987), no. 2, 359–387. MR 910206, DOI 10.1007/BF01388710
- Eduardo Esteves and Marco Pacini, Semistable modifications of families of curves and compactified Jacobians, Ark. Mat. 54 (2016), no. 1, 55–83. MR 3475818, DOI 10.1007/s11512-015-0220-4
- Eduardo Esteves, Compactifying the relative Jacobian over families of reduced curves, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3045–3095. MR 1828599, DOI 10.1090/S0002-9947-01-02746-5
- T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, 93–109. MR 1960923, DOI 10.1307/mmj/1049832895
- Samuel Grushevsky and Dmitry Zakharov, The zero section of the universal semiabelian variety and the double ramification cycle, Duke Math. J. 163 (2014), no. 5, 953–982. MR 3189435, DOI 10.1215/00127094-26444575
- Robin Hartshorne, Generalized divisors on Gorenstein schemes, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 287–339. MR 1291023, DOI 10.1007/BF00960866
- Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88. With an appendix by William Fulton. MR 664324, DOI 10.1007/BF01393371
- David Holmes, Extending the double ramification cycle by resolving the Abel–Jacobi map, arXiv:1707.02261, 2017.
- Alexis Kouvidakis, The Picard group of the universal Picard varieties over the moduli space of curves, J. Differential Geom. 34 (1991), no. 3, 839–850. MR 1139648
- Jesse Leo Kass and Nicola Pagani, Extensions of the universal theta divisor, Adv. Math. 321 (2017), 221–268. MR 3715710, DOI 10.1016/j.aim.2017.09.021
- Alex Massarenti, On the biregular geometry of the Fulton-MacPherson compactification, Adv. Math. 322 (2017), 97–131. MR 3720795, DOI 10.1016/j.aim.2017.10.012
- Margarida Melo, Compactified Picard stacks over $\overline {\scr M}_g$, Math. Z. 263 (2009), no. 4, 939–957. MR 2551606, DOI 10.1007/s00209-008-0447-x
- Margarida Melo, Compactified Picard stacks over the moduli stack of stable curves with marked points, Adv. Math. 226 (2011), no. 1, 727–763. MR 2735773, DOI 10.1016/j.aim.2010.07.012
- Margarida Melo, Compactifications of the universal Jacobian over curves with marked points, https://arxiv.org/abs/1509.06177v3, 2016.
- Nicole Mestrano, Conjecture de Franchetta forte, Invent. Math. 87 (1987), no. 2, 365–376 (French). MR 870734, DOI 10.1007/BF01389421
- J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR 861976
- Steffen Marcus and Jonathan Wise, Logarithmic compactification of the Abel–Jacobi section, arXiv:1708.04471, 2017.
- Yukihiko Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. I, Math. Ann. 221 (1976), no. 2, 97–141. MR 480537, DOI 10.1007/BF01433145
- Tadao Oda and C. S. Seshadri, Compactifications of the generalized Jacobian variety, Trans. Amer. Math. Soc. 253 (1979), 1–90. MR 536936, DOI 10.1090/S0002-9947-1979-0536936-4
- Rahul Pandharipande, A compactification over $\overline {M}_g$ of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), no. 2, 425–471. MR 1308406, DOI 10.1090/S0894-0347-96-00173-7
- Carlos T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47–129. MR 1307297
- S. Shadrin, M. Shapiro, and A. Vainshtein, Chamber behavior of double Hurwitz numbers in genus 0, Adv. Math. 217 (2008), no. 1, 79–96. MR 2357323, DOI 10.1016/j.aim.2007.06.016
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
- Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
- Kenji Ueno, On algebraic fibre spaces of abelian varieties, Math. Ann. 237 (1978), no. 1, 1–22. MR 506652, DOI 10.1007/BF01351555
- Eckart Viehweg, Canonical divisors and the additivity of the Kodaira dimension for morphisms of relative dimension one, Compositio Math. 35 (1977), no. 2, 197–223. MR 569690
Additional Information
- Jesse Leo Kass
- Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208
- MR Author ID: 990616
- Email: kassj@math.sc.edu
- Nicola Pagani
- Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, United Kingdom
- MR Author ID: 949513
- Email: pagani@liv.ac.uk
- Received by editor(s): May 17, 2018
- Received by editor(s) in revised form: October 5, 2018, and October 12, 2018
- Published electronically: June 21, 2019
- Additional Notes: The first author was supported by a grant from the Simons Foundation (Award Number 429929) and by the National Security Agency under Grant Number H98230-15-1-0264. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.
The second author was supported by the EPSRC First Grant EP/P004881/1 with the title “Wall-crossing on universal compactified Jacobians”. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4851-4887
- MSC (2010): Primary 14H10, 14H40, 14K10
- DOI: https://doi.org/10.1090/tran/7724
- MathSciNet review: 4009442