# zbMATH — the first resource for mathematics

Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups. (English) Zbl 1329.14051
The main object of the paper under review is a del Pezzo surface $$X$$ of degree 4 defined over a global field $$k$$ of characteristic different from 2. The authors consider the Brauer group of $$X$$ and prove that if $$X$$ is locally solvable (i.e., $$X(A_k)\neq\emptyset$$), then for every $$\mathcal A\in \mathrm{Br}(X)$$ there exists a projection $$g: X\dasharrow \mathbb P^1$$ (depending on $$\mathcal A$$), with at most two geometrically reducible fibres, such that $$\mathcal A\in g^*(\mathrm{Br} k(\mathbb P^1))$$. (In common slang, this means that $$\mathrm{Br}(X)$$ is vertical with respect to $$g$$.) Moreover, they prove that under the same conditions, there exists $$f: X\dasharrow \mathbb P^n$$ (where $$n=\dim_{\mathbb F_2}\mathrm{Br}(X)/\text{{im}} \mathrm{Br} (k)[2]$$) such that $$\mathrm{Br}(X)$$ is vertical with respect to $$f$$.
This allows one to deduce that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for certain quartic surfaces (under standard assumptions on the finiteness of the Tate-Shafarevich groups and Schinzel’s hypothesis).
On their way to the proof, the authors produce a practical algorithm for explicit computation of the Brauer group (which does not require computing the Galois action on exceptional curves).

##### MSC:
 14G05 Rational points 14F22 Brauer groups of schemes
Magma
Full Text:
##### References:
 [1] Bender, A. O.; Swinnerton-Dyer, P., Solubility of certain pencils of curves of genus 1, and of the intersection of two quadrics in $$\mathbb{P}^4$$, Proc. London Math. Soc. (3), 83, 2, 299-329, (2001), MR 1839456 (2002e:14033) · Zbl 1018.11031 [2] Birch, B. J.; Swinnerton-Dyer, H. P.F., The Hasse problem for rational surfaces, J. Reine Angew. Math., 274/275, 164-174, (1975), Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III, MR 0429913 (55 #2922) · Zbl 0326.14007 [3] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system. I. the user language, Computational Algebra and Number Theory, London, 1993, J. Symbolic Comput., 24, 3-4, 235-265, (1997), MR 1484478 · Zbl 0898.68039 [4] Bright, M., Efficient evaluation of the Brauer-Manin obstruction, Math. Proc. Cambridge Philos. Soc., 142, 1, 13-23, (2007), MR 2296387 (2007k:14026) · Zbl 1114.14010 [5] Bright, M. J.; Bruin, N.; Flynn, E. V.; Logan, A., The Brauer-Manin obstruction and $$\operatorname{Sh} [2]$$, LMS J. Comput. Math., 10, 354-377, (2007), (electronic), MR 2342713 (2008i:11087) · Zbl 1222.11084 [6] Colliot-Thélène, J.-L., Hasse principle for pencils of curves of genus one whose Jacobians have a rational 2-division point, close variation on a paper of Bender and Swinnerton-Dyer, (Rational Points on Algebraic Varieties, Progr. Math., vol. 199, (2001), Birkhäuser Basel), 117-161, (in English, with French summary), MR 1875172 (2003f:14017) · Zbl 1079.14510 [7] Colliot-Thélène, J.-L.; Poonen, B., Algebraic families of nonzero elements of Shafarevich-Tate groups, J. Amer. Math. Soc., 13, 1, 83-99, (2000), MR 1697093 (2000f:11067) · Zbl 0951.11022 [8] Colliot-Thélène, J.-L.; Sansuc, J.-J., La descente sur LES variétés rationnelles, (Journées de Géométrie Algébrique d’Angers, Juillet 1979, (1980), Sijthoff & Noordhoff Alphen aan den Rijn), 223-237, (in French), MR 605344 (82d:14016) · Zbl 0451.14018 [9] Colliot-Thélène, J.-L.; Sansuc, J.-J., Sur le principe de Hasse et l’approximation faible, et sur une hypothèse de schinzel, Acta Arith., 41, 1, 33-53, (1982), (in French), MR 667708 (83j:10019) · Zbl 0414.10009 [10] Colliot-Thélène, J.-L.; Skorobogatov, A. N., Good reduction of the Brauer-Manin obstruction, Trans. Amer. Math. Soc., 365, 2, 579-590, (2013) · Zbl 1317.14046 [11] Colliot-Thélène, J.-L.; Swinnerton-Dyer, P., Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. Reine Angew. Math., 453, 49-112, (1994), MR 1285781 (95h:11060) · Zbl 0805.14010 [12] Colliot-Thélène, J.-L.; Sansuc, J.-J.; Swinnerton-Dyer, P., Intersections of two quadrics and châtelet surfaces. I, J. Reine Angew. Math., 373, 37-107, (1987), MR 870307 (88m:11045a) · Zbl 0622.14029 [13] Colliot-Thélène, J.-L.; Skorobogatov, A.; Swinnerton-Dyer, P., Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math., 134, 3, 579-650, (1998), MR 1660925 (99k:11095) · Zbl 0924.14011 [14] Colliot-Thélène, J.-L.; Skorobogatov, A.; Swinnerton-Dyer, P., Rational points and zero-cycles on fibred varieties: schinzel’s hypothesis and Salberger’s device, J. Reine Angew. Math., 495, 1-28, (1998), MR 1603908 (99i:14027) · Zbl 0883.11029 [15] Colliot-Thélène, J.-L.; Harari, D.; Skorobogatov, A., Valeurs d’un polynôme à une variable représentées par une norme, (Number Theory and Algebraic Geometry, London Math. Soc. Lecture Note Ser., vol. 303, (2003), Cambridge University Press Cambridge), 69-89, (in French, with English summary), MR 2053456 (2005d:11095) · Zbl 1087.14016 [16] Grothendieck, A., Le groupe de Brauer. III. exemples et compléments, (Dix Exposés sur la Cohomologie des Schémas, (1968), North-Holland Amsterdam), 88-188, (in French), MR 0244271 (39 #5586c) · Zbl 0198.25901 [17] Harpaz, Y.; Skorobogatov, A.; Wittenberg, O., The Hardy-Littlewood conjecture and rational points, preprint available at · Zbl 1318.14024 [18] Kresch, A.; Tschinkel, Y., Effectivity of Brauer-Manin obstructions, Adv. Math., 218, 1, 1-27, (2008), MR 2409407 (2009e:14038) · Zbl 1142.14013 [19] Kunyavskiĭ, B.È.; Skorobogatov, A. N.; Tsfasman, M. A., Del Pezzo surfaces of degree four, Mém. Soc. Math. Fr. (N.S.), 37, 113, (1989), (in English, with French summary), MR 1016354 (90k:14035) · Zbl 0705.14039 [20] Manin, Yu. I., Le groupe de Brauer-Grothendieck en géométrie diophantienne, (Actes du Congrès International des Mathématiciens, Nice, 1970, (1971), Gauthier-Villars Paris), 401-411, MR 0427322 (55 #356) [21] Manin, Yu. I., Cubic forms: algebra, geometry, arithmetic, North-Holland Math. Library, vol. 4, (1974), North-Holland Publishing Co. Amsterdam, translated from Russian by M. Hazewinkel, MR 0460349 (57 #343) [22] Salberger, P.; Skorobogatov, A. N., Weak approximation for surfaces defined by two quadratic forms, Duke Math. J., 63, 2, 517-536, (1991), MR 1115119 (93e:11079) · Zbl 0770.14019 [23] Skorobogatov, A. N., Descent on fibrations over the projective line, Amer. J. Math., 118, 5, 905-923, (1996), MR 1408492 (97k:11099) · Zbl 0880.14008 [24] Skorobogatov, A., Torsors and rational points, Cambridge Tracts in Math., vol. 144, (2001), Cambridge University Press Cambridge, MR 1845760 (2002d:14032) · Zbl 0972.14015 [25] Swinnerton-Dyer, P., The Brauer group of cubic surfaces, Math. Proc. Cambridge Philos. Soc., 113, 3, 449-460, (1993), MR 1207510 (94a:14038) · Zbl 0804.14018 [26] Swinnerton-Dyer, P., Rational points on certain intersections of two quadrics, (Abelian Varieties, Egloffstein, 1993, (1995), de Gruyter Berlin), 273-292, MR 1336612 (97a:11099) · Zbl 0849.14009 [27] Swinnerton-Dyer, P., Brauer-Manin obstructions on some del Pezzo surfaces, Math. Proc. Cambridge Philos. Soc., 125, 2, 193-198, (1999), MR 1643855 (99h:11071) · Zbl 0951.14010 [28] Uematsu, T., On the Brauer group of diagonal cubic surfaces, Q. J. Math., (2013), in press [29] Várilly-Alvarado, A., Weak approximation on del Pezzo surfaces of degree 1, Adv. Math., 219, 6, 2123-2145, (2008), MR 2456278 (2009j:14045) · Zbl 1156.14017 [30] Wittenberg, O., Intersections de deux quadriques et pinceaux de courbes de genre 1, Lecture Notes in Math., vol. 1901, (2007), Springer Berlin, (in French), MR 2307807 (2008b:14029) · Zbl 1122.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.