On the Tate conjecture for the Fano surfaces of cubic threefolds. (English) Zbl 1285.14021

Roulleau, Xavier

J. Number Theory 133, No. 7, 2320-2323 (2013).

Summary: A Fano surface of a smooth cubic threefold \(X\hookrightarrow\mathbb P^4\) parametrizes the lines on \(X\). In this note, we prove that a Fano surface satisfies the Tate conjecture over a field of finite type over the prime field and characteristic not 2.

MSC:

14F20	Étale and other Grothendieck topologies and (co)homologies
14F30	\(p\)-adic cohomology, crystalline cohomology
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14C25	Algebraic cycles
14J45	Fano varieties

Keywords:

Tate conjecture; surface of general type; Fano surfaces of lines

References:

[1]	J. Achter, Arithmetic Torelli maps for cubic surfaces and threefolds, Trans. Amer. Math. Soc., in press, arXiv:1005.2131v3 [math.AG], preprint. · Zbl 1325.14053
[2]	Altman, A.; Kleiman, S., Foundations of the theory of Fano schemes, Compos. Math., 34, 1, 3-47, (1977) · Zbl 0414.14024
[3]	Beauville, A., LES singularités du diviseur θ de la jacobienne intermédiaire de lʼhypersurface cubique dans \(\mathbb{P}^4\), (Algebraic Threefolds, Varenna, 1981, Lecture Notes in Math., vol. 947, (1982), Springer Berlin, New York), 190-208
[4]	Bombieri, E.; Swinnerton-Dyer, H. P.F., On the local zeta function of a cubic threefold, Ann. Sc. Norm. Super. Pisa (3), 21, 1-29, (1967) · Zbl 0153.50501
[5]	Deligne, P., Cohomologie étale, (Boutot, J. F.; Grothendieck, A.; Illusie, L.; Verdier, J. L., SGA \(4 + 1 / 2\), Lecture Notes in Math., vol. 569, (1977), Springer-Verlag)
[6]	Deligne, P., LES conjectures de Weil II, Publ. Math. IHES, 52, 137-252, (1980) · Zbl 0456.14014
[7]	Faltings, G., Endlichkeitssätze für abelsche varietäten über zahlkörpern, Invent. Math., 73, 3, 349-366, (1983) · Zbl 0588.14026
[8]	Milne, J., Etale cohomology, Princeton Math. Ser., vol. 33, (1980), Princeton University Press Princeton, NJ, xiii+323 pp · Zbl 0433.14012
[9]	Mori, S., On Tate conjecture concerning endomorphisms of abelian varieties, (Proceedings of the International Symposium on Algebraic Geometry, (1977), Kyoto Univ. Kyoto), 219-230
[10]	Murre, J. P., Algebraic equivalence modulo rational equivalence on a cubic threefold, Compos. Math., 25, 161-206, (1972) · Zbl 0242.14002
[11]	Tate, J., Algebraic cycles and poles of zeta functions, (Proc. Conf. Purdue Univ., Arithmetical Algebraic Geom., vol. 965, (1963), Harper and Row New York), 93-110
[12]	Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 134-144, (1966) · Zbl 0147.20303
[13]	Tate, J., Conjectures on algebraic cycles in ℓ-adic cohomology, (Motives, Proc. Sympos. Pure Math., 55, Part 1, Seattle, WA, 1991, (1994), Amer. Math. Soc. Providence, RI), 71-83 · Zbl 0814.14009
[14]	Zarhin, Ju. G., Endomorphisms of abelian varieties over fields of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat., 39, 272-277, (1975), 471 (in Russian)