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Fano surfaces with 12 or 30 elliptic curves. (English) Zbl 1225.14029
The author studies the properties of Fano surfaces with a fixed number of elliptic curves. We recall that a Fano surface is a surface of general type parametrizing the lines of a smooth cubic threefold.
The paper gives an application of the results of a previous work [Manuscr. Math. 129, No. 3, 381–399 (2009; Zbl 1177.14079)], to the case of surfaces having 12 or 30 elliptic curves, describing the first ones as ramified triple covers of an abelian surface and computing their Néron-Severi group. In the latter case, the author focuses on the Fermat cubic, explicitly computing the configuration of the 30 elliptic curves and the generators of the Néron-Severi group.

MSC:
14J29 Surfaces of general type
14J50 Automorphisms of surfaces and higher-dimensional varieties
14C20 Divisors, linear systems, invertible sheaves
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[1] W. Barth, K. Hulek, C. Peters, and A. Van De Ven, Compact complex surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer-Verlag, Berlin, 2004. · Zbl 1036.14016
[2] A. Beauville, Les singularités du diviseur \(\Theta\) de la jacobienne intermédiaire de la cubique dans \(\mathbb P^4,\) Algebraic threefolds (Varenna, 1981), Lecture Notes in Math., 947, pp. 190-208, Springer-Verlag, Berlin, 1982. · Zbl 0492.14033
[3] C. Birkenhake and H. Lange, Complex Abelian varieties, 2nd ed., Grundlehren Math. Wiss., 302, Springer-Verlag, Berlin, 2004. · Zbl 1056.14063
[4] E. Bombieri and H. P. F. Swinnerton-Dyer, On the local Zeta function of a cubic threefold, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 1-29. · Zbl 0153.50501 · numdam:ASNSP_1967_3_21_1_1_0 · eudml:83410
[5] H. Clemens and P. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281-356. · Zbl 0214.48302 · doi:10.2307/1970801
[6] A. Collino, The fundamental group of the Fano surface I, II, Algebraic threefolds (Varenna, 1981), Lecture Notes in Math, 947, pp. 209-218, 219-220, Springer-Verlag, New York, 1982.
[7] F. Gherardelli, Un “osservazione sulla varieta” cubica di \(\mathbb P^4,\) Rend. Sem. Mat. Fis. Milano 37 (1967), 157-160. · Zbl 0199.55902 · doi:10.1007/BF02925643
[8] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. · Zbl 0408.14001
[9] S. Mori, The endomorphism rings of some Abelian varieties, Japan. J. Math. (N.S.) 2 (1976), 109-130. · Zbl 0339.14016
[10] J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math. 25 (1972), 161-206. · Zbl 0242.14002 · numdam:CM_1972__25_2_161_0 · eudml:89140
[11] V. Nikulin, On Kummer surfaces, Math. USSR-Izv. 9 (1975), 261-275. · Zbl 0325.14015 · doi:10.1070/IM1975v009n02ABEH001477
[12] X. Roulleau, Elliptic curve configurations on Fano surfaces, Manuscripta Math. 129 (2009), 381-399. · Zbl 1177.14079 · doi:10.1007/s00229-009-0264-5
[13] —, L’application cotangente des surfaces de type général, Geom. Dedicata 142 (2009), 151-171. · Zbl 1180.14041 · doi:10.1007/s10711-009-9364-3
[14] A. N. Tyurin, On the Fano surface of a nonsingular cubic in \(\mathbb P^4,\) Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 1200-1208. · Zbl 0205.25403
[15] —, The geometry of the Fano surface of a nonsingular cubic \(F\subset\mathbb P^4\) and Torelli theorems for Fano surfaces and cubics, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 498-529. · Zbl 0215.08201
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