We prove that the geometric genus of a curve in a very generic Jacobian of dimension satisfies either or . This gives a positive answer to a conjecture of Naranjo and Pirola. For small values of the second inequality can be further improved to .
@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}, pages = {735--754}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 12}, number = {3}, year = {2013}, zbl = {1300.14033}, mrnumber = {3137462}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2013_5_12_3_735_0/} }
TY - JOUR AU - Marcucci, Valeria Ornella TI - On the genus of curves in a Jacobian variety JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2013 DA - 2013/// SP - 735 EP - 754 VL - Ser. 5, 12 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2013_5_12_3_735_0/ UR - https://zbmath.org/?q=an%3A1300.14033 UR - https://www.ams.org/mathscinet-getitem?mr=3137462 LA - en ID - ASNSP_2013_5_12_3_735_0 ER -
Marcucci, Valeria Ornella. On the genus of curves in a Jacobian variety. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 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. | EuDML 139854 | 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. | EuDML 90359 | 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 lying on a -dimensional Jacobian variety, Invent. Math. 95 (1989), 263–276. | EuDML 143653 | 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. | EuDML 164467 | 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. | EuDML 153453 | 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