Prym varieties of genus four curves

Bruin, Nils; Sertöz, Emre

doi:10.1090/tran/7902

Prym varieties of genus four curves

Authors: Nils Bruin and Emre Can Sertöz
Journal: Trans. Amer. Math. Soc. 373 (2020), 149-183
MSC (2010): Primary 14H45, 14H40, 14H50
DOI: https://doi.org/10.1090/tran/7902
Published electronically: September 23, 2019
MathSciNet review: 4042871
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Double covers of a generic genus four curve C are in bijection with Cayley cubics containing the canonical model of C. The Prym variety associated to a double cover is a quadratic twist of the Jacobian of a genus three curve X. The curve X can be obtained by intersecting the dual of the corresponding Cayley cubic with the dual of the quadric containing C. We take this construction to its limit, studying all smooth degenerations and proving that the construction, with appropriate modifications, extends to the complement of a specific divisor in moduli. We work over an arbitrary field of characteristic different from two in order to facilitate arithmetic applications.

[1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
[2] Arnaud Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786479, https://doi.org/10.1307/mmj/1030132707
[3] Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR 1217487
[4] Nils Bruin, The arithmetic of Prym varieties in genus 3, Compos. Math. 144 (2008), no. 2, 317–338. MR 2406115, https://doi.org/10.1112/S0010437X07003314
[5] Nils Bruin and E. Victor Flynn, Towers of 2-covers of hyperelliptic curves, Trans. Amer. Math. Soc. 357 (2005), no. 11, 4329–4347. MR 2156713, https://doi.org/10.1090/S0002-9947-05-03954-1
[6] N. Bruin and E. V. Flynn, Rational divisors in rational divisor classes, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 132–139. MR 2137349, https://doi.org/10.1007/978-3-540-24847-7_9
[7] F. Catanese, On the rationality of certain moduli spaces related to curves of genus 4, Algebraic geometry (Ann Arbor, Mich., 1981) Lecture Notes in Math., vol. 1008, Springer, Berlin, 1983, pp. 30–50. MR 723706, https://doi.org/10.1007/BFb0065697
[8] F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465. MR 620679, https://doi.org/10.1007/BF01389064
[9] Arthur B. Coble, Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, vol. 10, American Mathematical Society, Providence, R.I., 1982. Reprint of the 1929 edition. MR 733252
[10] Andrea Del Centina and Sevín Recillas, Some projective geometry associated with unramified double covers of curves of genus 4, Ann. Mat. Pura Appl. (4) 133 (1983), 125–140 (English, with Italian summary). MR 725022, https://doi.org/10.1007/BF01766014
[11] Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027
[12] 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, https://doi.org/10.1090/conm/136/1188194
[13] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
[14] Laura Hidalgo-Solís and Sevin Recillas-Pishmish, The fibre of the Prym map in genus four, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 2 (1999), no. 1, 219–229 (English, with Italian summary). MR 1794551
[15] E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann. 284 (1989), no. 2, 307–327. MR 1000113, https://doi.org/10.1007/BF01442878
[16] W. P. Milne, Sextactic Cones and Tritangent Planes of the Same System of a Quadri-Cubic Curve, Proc. London Math. Soc. (2) 21 (1923), 373–380. MR 1575365, https://doi.org/10.1112/plms/s2-21.1.373
[17] 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
[18] Sevin Recillas, Jacobians of curves with 𝑔¹₄’s are the Prym’s of trigonal curves, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 1, 9–13. MR 480505
[19] Sevin Recillas, Symmetric cubic surfaces and curves of genus 3 and 4, Boll. Un. Mat. Ital. B (7) 7 (1993), no. 4, 787–819 (English, with Italian summary). MR 1255648
[20] Paul Roth, Über Beziehungen zwischen algebraischen Gebilden vom Geschlechte drei und vier, Monatsh. Math. Phys. 22 (1911), no. 1, 64–88 (German). MR 1548122, https://doi.org/10.1007/BF01742792
[21] E. A. Tevelev, Projective duality and homogeneous spaces, Encyclopaedia of Mathematical Sciences, vol. 133, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, IV. MR 2113135
[22] Ravi Vakil, Twelve points on the projective line, branched covers, and rational elliptic fibrations, Math. Ann. 320 (2001), no. 1, 33–54. MR 1835061, https://doi.org/10.1007/PL00004469