Compactifying moduli spaces
Author:
Lucia Caporaso
Journal:
Bull. Amer. Math. Soc. 57 (2020), 455-482
MSC (2010):
Primary 14H10, 14D06, 14D20
DOI:
https://doi.org/10.1090/bull/1662
Published electronically:
December 17, 2018
MathSciNet review:
4108092
Full-text PDF Free Access
View in AMS MathViewer
Abstract | References | Similar Articles | Additional Information
Abstract: The boundary of some well known algebro-geometric moduli spaces is described by highlighting the recursive combinatorial properties in connection with tropical and non-Archimedean geometry.
- Dan Abramovich, Lucia Caporaso, and Sam Payne, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 4, 765–809 (English, with English and French summaries). MR 3377065, DOI https://doi.org/10.24033/asens.2258
- Valery Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (2002), no. 3, 611–708. MR 1923963, DOI https://doi.org/10.2307/3062130
- Enrico Arbarello, Maurizio Cornalba, and Phillip A. Griffiths, Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 268, Springer, Heidelberg, 2011. With a contribution by Joseph Daniel Harris. MR 2807457
- M. Artin, Néron models, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 213–230. MR 861977
- Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR 572974, DOI https://doi.org/10.1007/BF01418373
- Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
- Silvia Brannetti, Margarida Melo, and Filippo Viviani, On the tropical Torelli map, Adv. Math. 226 (2011), no. 3, 2546–2586. MR 2739784, DOI https://doi.org/10.1016/j.aim.2010.09.011
- 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 https://doi.org/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 https://doi.org/10.1353/ajm.2008.0000
- Lucia Caporaso and Karl Christ, Combinatorics of compactified universal Jacobians. arXiv:1801.04098 (2018).
- Lucia Caporaso and Filippo Viviani, Torelli theorem for stable curves, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 5, 1289–1329. MR 2825165, DOI https://doi.org/10.4171/JEMS/281
- Renzo Cavalieri, Hannah Markwig, and Dhruv Ranganathan, Tropicalizing the space of admissible covers, Math. Ann. 364 (2016), no. 3-4, 1275–1313. MR 3466867, DOI https://doi.org/10.1007/s00208-015-1250-8
- Karl Christ, Orientations, break Divisors and compactified Jacobians, PhD thesis in Mathematics, Roma Tre University (2017).
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
- D. Gieseker, Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York, 1982. MR 691308
- Alexander Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique. V. Les schémas de Picard: théorèmes d’existence, Séminaire Bourbaki, Vol. 7, Soc. Math. France, Paris, 1995, pp. Exp. No. 232, 143–161 (French). MR 1611170
- Alexander Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249–276 (French). MR 1611822
- Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
- 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 https://doi.org/10.1007/BF01393371
- Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$, Math. Scand. 52 (1983), no. 2, 161–199. MR 702953, DOI https://doi.org/10.7146/math.scand.a-12001
- Grigory Mikhalkin, Moduli spaces of rational tropical curves, Proceedings of Gökova Geometry-Topology Conference 2006, Gökova Geometry/Topology Conference (GGT), Gökova, 2007, pp. 39–51. MR 2404949
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI https://doi.org/10.1007/bf02684271
- Tadao Oda and C. S. Seshadri, Compactifications of the generalized Jacobian variety, Trans. Amer. Math. Soc. 253 (1979), 1–90. MR 536936, DOI https://doi.org/10.1090/S0002-9947-1979-0536936-4
- M. Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 27–76 (French). MR 282993
- Amaury Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscripta Math. 123 (2007), no. 4, 381–451 (French, with English summary). MR 2320738, DOI https://doi.org/10.1007/s00229-007-0094-2
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
- Filippo Viviani, Tropicalizing vs. compactifying the Torelli morphism, Tropical and non-Archimedean geometry, Contemp. Math., vol. 605, Amer. Math. Soc., Providence, RI, 2013, pp. 181–210. MR 3204272, DOI https://doi.org/10.1090/conm/605/12116
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Additional Information
Lucia Caporaso
Affiliation:
Dipartimento di Matematica e Fisica, Università Roma Tre, Largo San Leonardo Murialdo, I-00146 Roma, Italy
MR Author ID:
345125
Email:
caporaso@mat.uniroma3.it
Received by editor(s):
June 15, 2018
Published electronically:
December 17, 2018
Article copyright:
© Copyright 2018
American Mathematical Society