On the genus of curves in a Jacobian variety
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, p. 735-754

We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For small values of g the second inequality can be further improved to p>2g-2.

Published online : 2019-02-21
Classification:  14H40,  32G20 
@article{ASNSP_2013_5_12_3_735_0,
     author = {Marcucci, Valeria Ornella},
     title = {On the genus of curves in a Jacobian variety},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {3},
     year = {2013},
     pages = {735-754},
     zbl = {1300.14033},
     mrnumber = {3137462},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_3_735_0}
}

Marcucci, Valeria Ornella. On the genus of curves in a Jacobian variety. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 3, pp. 735-754. http://www.numdam.org/item/ASNSP_2013_5_12_3_735_0/

[1] E. Arbarello and M. Cornalba, On a conjecture of Petri, Comment. Math. Helv. 56 (1981), 1–38. | MR 615613 | Zbl 0505.14002

[2] V. Alexeev, Compactified Jacobians and Torelli map, Publ. Res. Inst. Math. Sci. 40 (2004), 1241–1265. | MR 2105707 | Zbl 1079.14019

[3] A. Andreotti, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801–828. | MR 102518 | Zbl 0084.17304

[4] F. Bardelli, C. Ciliberto and A. Verra, Curves of minimal genus on a general Abelian variety, Compos. Math. 96 (1995), 115–147. | Numdam | MR 1326709 | Zbl 0864.14027

[5] C. Birkenhake and H. Lange, “Complex Abelian Varieties”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 302, Springer-Verlag, Berlin, second edition, 2004. | MR 2062673 | Zbl 1056.14063

[6] S. Bosch, W. Lütkebohmert and M. 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 | Zbl 0705.14001

[7] F. Bardelli and G. P. Pirola, Curves of genus g lying on a g-dimensional Jacobian variety, Invent. Math. 95 (1989), 263–276. | MR 974904 | Zbl 0638.14025

[8] C.-L. Chai, “Compactification of Siegel Moduli Schemes” London Mathematical Society Lecture Note Series, Vol. 107, Cambridge University Press, Cambridge, 1985. | MR 853543 | Zbl 0578.14009

[9] C. Ciliberto, G. van der Geer and M. Teixidor i Bigas, On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms, J. Algebr. Geom. 1 (1992), 215–229. | MR 1144437 | Zbl 0806.14020

[10] G. Faltings and C.-L. Chai, “Degeneration of Abelian Varieties”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 22, with an appendix by David Mumford, Springer-Verlag, Berlin, 1990. | MR 1083353 | Zbl 0744.14031

[11] R. Hartshorne, “Algebraic Geometry”, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. | MR 463157 | Zbl 0367.14001

[12] V. Marcucci and G. P. Pirola, Generic Torelli theorem for Prym varieties of ramified coverings, Compos. Math. 148 (2012), 1147–1170. | MR 2956039 | Zbl 1254.14033

[13] J. C. Naranjo and G. P. Pirola, On the genus of curves in the generic Prym variety, Indag. Math. (N.S.) 5 (1994), 101–105. | MR 1268732 | Zbl 0823.14033

[14] T. Oda and C. S. Seshadri, Compactifications of the generalized Jacobian variety, Trans. Amer. Math. Soc. 253 (1979), 1–90. | MR 536936 | Zbl 0418.14019

[15] G. P. Pirola, Base number theorem for Abelian varieties. An infinitesimal approach, Math. Ann. 282 (1988), 361–368. | MR 967018 | Zbl 0625.14024

[16] G. P. Pirola, Curves on generic Kummer varieties, Duke Math. J. 59 (1989), 701–708. | MR 1046744 | Zbl 0717.14021

[17] G. P. Pirola, On a conjecture of Xiao, J. Reine Angew. Math. 431 (1992), 75–89. | MR 1179333 | Zbl 0753.14040

[18] J.-P. Serre, “Algebraic Groups and Class Fields”, Graduate Texts in Mathematics, Vol. 117, translated from the French, Springer-Verlag, New York, 1988. | MR 918564 | Zbl 0703.14001

[19] C. Voisin, “Hodge Theory and Complex Algebraic Geometry. II”, Cambridge Studies in Advanced Mathematics, Vol. 77, translated from the French by Leila Schneps, Cambridge University Press, Cambridge, 2003. | MR 1997577 | Zbl 1032.14002