Prym varieties of genus four curves

Bruin, Nils; Sertöz, Emre

doi:10.1090/tran/7902

Prym varieties of genus four curves
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by Nils Bruin and Emre Can Sertöz PDF

Trans. Amer. Math. Soc. 373 (2020), 149-183 Request permission

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.

References

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, DOI 10.1007/978-1-4757-5323-3
Arnaud Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786479, DOI 10.1307/mmj/1030132707
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, DOI 10.1007/978-3-662-02788-2
Nils Bruin, The arithmetic of Prym varieties in genus 3, Compos. Math. 144 (2008), no. 2, 317–338. MR 2406115, DOI 10.1112/S0010437X07003314
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, DOI 10.1090/S0002-9947-05-03954-1
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, DOI 10.1007/978-3-540-24847-7_{9}
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, DOI 10.1007/BFb0065697
F. Catanese, Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications, Invent. Math. 63 (1981), no. 3, 433–465. MR 620679, DOI 10.1007/BF01389064
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
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, DOI 10.1007/BF01766014
Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
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
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
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
E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann. 284 (1989), no. 2, 307–327. MR 1000113, DOI 10.1007/BF01442878
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, DOI 10.1112/plms/s2-21.1.373
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
Sevin Recillas, Jacobians of curves with $g^{1}_{4}$’s are the Prym’s of trigonal curves, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 1, 9–13. MR 480505
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
Paul Roth, Über Beziehungen zwischen algebraischen Gebilden vom Geschlechte drei und vier, Monatsh. Math. Phys. 22 (1911), no. 1, 64–88 (German). MR 1548122, DOI 10.1007/BF01742792
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
Ravi Vakil, Twelve points on the projective line, branched covers, and rational elliptic fibrations, Math. Ann. 320 (2001), no. 1, 33–54. MR 1835061, DOI 10.1007/PL00004469