Global Prym-Torelli for double coverings ramified in at least six points
Authors:
Juan Carlos Naranjo and Angela Ortega
Journal:
J. Algebraic Geom. 31 (2022), 387-396
DOI:
https://doi.org/10.1090/jag/779
Published electronically:
December 28, 2021
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove that the ramified Prym map $\mathcal P_{g, r}$ which sends a covering $\pi :D\longrightarrow C$ ramified in $r$ points to the Prym variety $P(\pi )≔Ker(Nm_{\pi })$ is an embedding for all $r\ge 6$ and for all $g(C)>0$. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that $\mathcal P_{g, 2}$ and $\mathcal P_{g, 4}$ have positive dimensional fibers.
References
- Fabio Bardelli, Ciro Ciliberto, and Alessandro Verra, Curves of minimal genus on a general abelian variety, Compositio Math. 96 (1995), no. 2, 115–147. MR 1326709
- Christina Birkenhake and Herbert Lange, An isomorphism between moduli spaces of abelian varieties, Math. Nachr. 253 (2003), 3–7. MR 1976843, DOI 10.1002/mana.200310041
- Ron Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55–125. MR 1188194, DOI 10.1090/conm/136/1188194
- Ron Donagi, The tetragonal construction, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 181–185. MR 598683, DOI 10.1090/S0273-0979-1981-14875-8
- Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
- Atsushi Ikeda, Global Prym-Torelli theorem for double coverings of elliptic curves, Algebr. Geom. 7 (2020), no. 5, 544–560. MR 4156417, DOI 10.14231/ag-2020-019
- E. Izadi and H. Lange, Counter-examples of high Clifford index to Prym-Torelli, J. Algebraic Geom. 21 (2012), no. 4, 769–787. MR 2957696, DOI 10.1090/S1056-3911-2012-00587-6
- Herbert Lange and Angela Ortega, Prym varieties of cyclic coverings, Geom. Dedicata 150 (2011), 391–403. MR 2753712, DOI 10.1007/s10711-010-9512-9
- H. Lange and A. Ortega, The trigonal construction in the ramified case, arXiv:1902.00251 (2019).
- Valeria Ornella Marcucci and Juan Carlos Naranjo, Prym varieties of double coverings of elliptic curves, Int. Math. Res. Not. IMRN 6 (2014), 1689–1698. MR 3180606, DOI 10.1093/imrn/rns271
- Valeria Ornella Marcucci and Gian Pietro Pirola, Generic Torelli theorem for Prym varieties of ramified coverings, Compos. Math. 148 (2012), no. 4, 1147–1170. MR 2956039, DOI 10.1112/S0010437X12000280
- Henrik H. Martens, An extended Torelli theorem, Amer. J. Math. 87 (1965), 257–261. MR 182624, DOI 10.2307/2373002
- David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510
- D. S. Nagaraj and S. Ramanan, Polarisations of type $(1,2,\cdots ,2)$ on abelian varieties, Duke Math. J. 80 (1995), no. 1, 157–194. MR 1360615, DOI 10.1215/S0012-7094-95-08007-7
- Juan-Carlos Naranjo, The positive-dimensional fibres of the Prym map, Pacific J. Math. 172 (1996), no. 1, 223–226. MR 1379294, DOI 10.2140/pjm.1996.172.223
- Juan Carlos Naranjo and Angela Ortega, Generic injectivity of the Prym map for double ramified coverings, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3627–3646. With an appendix by Alessandro Verra. MR 3896124, DOI 10.1090/tran/7459
- Stefanos Pantazis, Prym varieties and the geodesic flow on $\textrm {SO}(n)$, Math. Ann. 273 (1986), no. 2, 297–315. MR 817884, DOI 10.1007/BF01451409
- Ziv Ran, On a theorem of Martens, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), no. 2, 287–291 (1987). MR 906019
References
- Fabio Bardelli, Ciro Ciliberto, and Alessandro Verra, Curves of minimal genus on a general abelian variety, Compositio Math. 96 (1995), no. 2, 115–147. MR 1326709
- Christina Birkenhake and Herbert Lange, An isomorphism between moduli spaces of abelian varieties, Math. Nachr. 253 (2003), 3–7. MR 1976843, DOI 10.1002/mana.200310041
- Ron Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55–125. MR 1188194, DOI 10.1090/conm/136/1188194
- Ron Donagi, The tetragonal construction, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 181–185. MR 598683, DOI 10.1090/S0273-0979-1981-14875-8
- Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
- Atsushi Ikeda, Global Prym-Torelli theorem for double coverings of elliptic curves, Algebr. Geom. 7 (2020), no. 5, 544–560. MR 4156417, DOI 10.14231/ag-2020-019
- E. Izadi and H. Lange, Counter-examples of high Clifford index to Prym-Torelli, J. Algebraic Geom. 21 (2012), no. 4, 769–787. MR 2957696, DOI 10.1090/S1056-3911-2012-00587-6
- Herbert Lange and Angela Ortega, Prym varieties of cyclic coverings, Geom. Dedicata 150 (2011), 391–403. MR 2753712, DOI 10.1007/s10711-010-9512-9
- H. Lange and A. Ortega, The trigonal construction in the ramified case, arXiv:1902.00251 (2019).
- Valeria Ornella Marcucci and Juan Carlos Naranjo, Prym varieties of double coverings of elliptic curves, Int. Math. Res. Not. IMRN 6 (2014), 1689–1698. MR 3180606, DOI 10.1093/imrn/rns271
- Valeria Ornella Marcucci and Gian Pietro Pirola, Generic Torelli theorem for Prym varieties of ramified coverings, Compos. Math. 148 (2012), no. 4, 1147–1170. MR 2956039, DOI 10.1112/S0010437X12000280
- Henrik H. Martens, An extended Torelli theorem, Amer. J. Math. 87 (1965), 257–261. MR 182624, DOI 10.2307/2373002
- David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510
- D. S. Nagaraj and S. Ramanan, Polarisations of type $(1,2,\cdots ,2)$ on abelian varieties, Duke Math. J. 80 (1995), no. 1, 157–194. MR 1360615, DOI 10.1215/S0012-7094-95-08007-7
- Juan-Carlos Naranjo, The positive-dimensional fibres of the Prym map, Pacific J. Math. 172 (1996), no. 1, 223–226. MR 1379294
- Juan Carlos Naranjo and Angela Ortega, Generic injectivity of the Prym map for double ramified coverings, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3627–3646. With an appendix by Alessandro Verra. MR 3896124, DOI 10.1090/tran/7459
- Stefanos Pantazis, Prym varieties and the geodesic flow on $\mathrm {SO}(n)$, Math. Ann. 273 (1986), no. 2, 297–315. MR 817884, DOI 10.1007/BF01451409
- Ziv Ran, On a theorem of Martens, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), no. 2, 287–291 (1987). MR 906019
Additional Information
Juan Carlos Naranjo
Affiliation:
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Spain
MR Author ID:
318430
ORCID:
0000-0003-1989-4924
Email:
jcnaranjo@ub.edu
Angela Ortega
Affiliation:
Institut für Mathematik, Humboldt Universität zu Berlin, Germany
MR Author ID:
726216
Email:
ortega@math.hu-berlin.de
Received by editor(s):
June 26, 2020
Received by editor(s) in revised form:
November 4, 2020
Published electronically:
December 28, 2021
Additional Notes:
The first author was partially supported by the Proyecto de Investigación MTM2015-65361-PID2019-104047GB-100.
Article copyright:
© Copyright 2021
University Press, Inc.
