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On the Severi problem in arbitrary characteristic
Publications Mathématiques de l'IHÉS, Volume 137 (2023), pp. 1-45
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corrected-by Correction: On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.

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DOI: 10.1007/s10240-022-00135-x

Karl Christ 1; Xiang He 1; Ilya Tyomkin 1

1 
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Karl Christ; Xiang He; Ilya Tyomkin. On the Severi problem in arbitrary characteristic. Publications Mathématiques de l'IHÉS, Volume 137 (2023), pp. 1-45. doi: 10.1007/s10240-022-00135-x

[ACGS20.] D. Abramovich; Q. Chen; M. Gross; B. Siebert Decomposition of degenerate Gromov-Witten invariants, Compos. Math., Volume 156 (2020), pp. 2020-2075 | MR | DOI | Zbl

[ACP15.] D. Abramovich; L. Caporaso; S. Payne The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4), Volume 48 (2015), pp. 765-809 | MR | Zbl | Numdam | DOI

[AC81a.] E. Arbarello; M. Cornalba Footnotes to a paper of Beniamino Segre, Math. Ann., Volume 256 (1981), pp. 341-362 | MR | DOI | Zbl

[AC81b.] E. Arbarello; M. Cornalba Su una congettura di Petri, Comment. Math. Helv., Volume 56 (1981), pp. 1-37 | MR | DOI | Zbl

[BIMS15.] E. Brugallé; I. Itenberg; G. Mikhalkin; K. Shaw Brief introduction to tropical geometry, Proceedings of the Gökova Geometry-Topology Conference 2014 (2015), pp. 1-75 | Zbl | MR

[BM09.] E. Brugallé; G. Mikhalkin Floor decompositions of tropical curves: the planar case, Proceedings of Gökova Geometry-Topology Conference 2008 (2009), pp. 64-90 | Zbl | MR

[BPR13.] M. Baker; S. Payne; J. Rabinoff On the structure of non-Archimedean analytic curves, Tropical and Non-Archimedean Geometry, 605, Am. Math. Soc., Providence, 2013, pp. 93-121 | DOI | Zbl | MR

[CC99.] L. Chiantini; C. Ciliberto On the Severi varieties on surfaces in 𝐏 3 , J. Algebraic Geom., Volume 8 (1999), pp. 67-83 | MR | Zbl

[CCUW20.] R. Cavalieri; M. Chan; M. Ulirsch; J. Wise A moduli stack of tropical curves, Forum Math. Sigma, Volume 8 (2020) | MR | DOI | Zbl

[CHL06.] H.-J. Chiang-Hsieh; J. Lipman A numerical criterion for simultaneous normalization, Duke Math. J., Volume 133 (2006), pp. 347-390 | MR | DOI | Zbl

[CHT22a.] K. Christ; X. He; I. Tyomkin Degeneration of curves on some polarized toric surfaces, J. Reine Angew. Math., Volume 787 (2022), pp. 197-240 | MR | DOI | Zbl

[CHT22b.] K. Christ, X. He and I. Tyomkin, Irreducibility of Severi varieties and Hurwitz schemes in small characteristic, 2022, in preparation.

[Del85.] P. Deligne Le lemme de Gabber, Astérisque, Volume 127 (1985), pp. 131-150 Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84) | MR | Zbl | Numdam

[dJ96.] A. J. de Jong Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci., Volume 83 (1996), pp. 51-93 | MR | Zbl | Numdam | DOI

[dJHS11.] A. J. de Jong; X. He; J. M. Starr Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, Publ. Math. Inst. Hautes Études Sci., Volume 114 (2011), pp. 1-85 | MR | DOI | Zbl | Numdam

[DM69.] P. Deligne; D. Mumford The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci., Volume 36 (1969), pp. 75-109 | MR | DOI | Zbl | Numdam

[DPT80.] Séminaire sur les Singularités des Surfaces (M. Demazure; H. C. Pinkham; B. Teissier, eds.), 777, Springer, Berlin, 1980, pp. 1976-1977 (held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau) | MR

[EC85.] F. Enriques; O. Chisini Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. 1. Vol. I, II, 5, Zanichelli Editore S.p.A., Bologna, 1985 (Reprint of the 1915 and 1918 editions) | Zbl

[Ful69.] W. Fulton Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. Math. (2), Volume 90 (1969), pp. 542-575 | MR | DOI | Zbl

[GM07.] A. Gathmann; H. Markwig The numbers of tropical plane curves through points in general position, J. Reine Angew. Math., Volume 602 (2007), pp. 155-177 | MR | Zbl

[Har86.] J. Harris On the Severi problem, Invent. Math., Volume 84 (1986), pp. 445-461 | MR | DOI | Zbl

[HM82.] J. Harris; D. Mumford On the Kodaira dimension of the moduli space of curves, Invent. Math., Volume 67 (1982), pp. 23-88 (with an appendix by William Fulton) | MR | DOI | Zbl

[Hur91.] A. Hurwitz Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891), pp. 1-60 | MR | DOI | Zbl

[Ill71.] L. Illusie Complexe cotangent et déformations. I, 239, Springer, Berlin, 1971 | Zbl | MR | DOI

[Kle82.] F. Klein Über Riemann’s Theorie der algebraischen Functionen und ihrer Integrale, Teubner, Leipzig, 1882 | Zbl

[Knu83.] F. F. Knudsen The projectivity of the moduli space of stable curves. III. The line bundles on M g,n , and a proof of the projectivity of M ¯ g,n in characteristic 0, Math. Scand., Volume 52 (1983), pp. 200-212 | MR | DOI | Zbl

[Kon95.] M. Kontsevich Enumeration of rational curves via torus actions, The Moduli Space of Curves, Volume 129 (1995), pp. 335-368 | Zbl | MR | DOI

[KS13.] S. L. Kleiman; V. V. Shende On the Göttsche threshold, A Celebration of Algebraic Geometry, 18, Am. Math. Soc., Providence, 2013, pp. 429-449 (with an appendix by Ilya Tyomkin) | MR | Zbl

[Lan20.] L. Lang Monodromy of rational curves on toric surfaces, J. Topol., Volume 13 (2020), pp. 1658-1681 | MR | DOI | Zbl

[Mik05.] G. Mikhalkin Enumerative tropical algebraic geometry in 𝐑 2 , J. Am. Math. Soc., Volume 18 (2005), pp. 313-377 | DOI | Zbl | MR

[Mor79.] S. Mori Projective manifolds with ample tangent bundles, Ann. Math. (2), Volume 110 (1979), pp. 593-606 | MR | DOI | Zbl

[Ran19.] D. Ranganathan Logarithmic Gromov-Witten theory with expansions, Algebr. Geom., Volume 9 (2022), pp. 714-761 | MR | DOI | Zbl

[Rau17.] J. Rau, A first expedition to tropical geometry, 2017, https://www.math.uni-tuebingen.de/user/jora/downloads/FirstExpedition.pdf.

[Ros52.] M. Rosenlicht Equivalence relations on algebraic curves, Ann. Math. (2), Volume 56 (1952), pp. 169-191 | MR | DOI | Zbl

[Sev21.] F. Severi Vorlesungen über algebraische Geometrie, Anhang F, Teubner, Leipzig, 1921 | DOI | Zbl

[Sta20.] The Stacks project authors, The stacks project, 2020, https://stacks.math.columbia.edu.

[Tyo07.] I. Tyomkin On Severi varieties on Hirzebruch surfaces, Int. Math. Res. Not., Volume 23 (2007) | MR | Zbl

[Tyo12.] I. Tyomkin Tropical geometry and correspondence theorems via toric stacks, Math. Ann., Volume 353 (2012), pp. 945-995 | MR | DOI | Zbl

[Tyo13.] I. Tyomkin On Zariski’s theorem in positive characteristic, J. Eur. Math. Soc., Volume 15 (2013), pp. 1783-1803 | MR | DOI | Zbl

[Tyo14.] I. Tyomkin An example of a reducible Severi variety, Proceedings of the Gökova Geometry-Topology Conference 2013 (2014), pp. 33-40 | Zbl | MR

[Zar82.] O. Zariski Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes, Am. J. Math., Volume 104 (1982), pp. 209-226 | MR | DOI | Zbl

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