The Brauer group of an affine double plane associated to a hyperelliptic curve

References

• Artin, M. (1962). Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84:485–496 [MR 0146182 (26 #3704)]
   Web of Science ®Google Scholar
• Artin, M. (1969). Algebraic approximation of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math. 36:23–58 [MR 0268188 (42 #3087)].
   Google Scholar
• Artin, M., Mumford, D. (1972). Some elementary examples of unirational varieties which are not rational. Some elementary examples of unirational varieties which are not rational s3-25(1):75–95 [MR 0321934 (48 #299)].
   Google Scholar
• Auslander, M., Goldman, O. (1960). The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97:367–409 [MR 0121392 (22 #12130)].
   Google Scholar
• Bourbaki, N. (1989). Commutative Algebra. Chapters 1–7. Elements of Mathematics (Berlin). Berlin: Springer-Verlag [Translated from the French, Reprint of the 1972 edition. MR 979760 (90a:13001)].
   Google Scholar
• Bright, M. (2013). Brauer groups of singular del Pezzo surfaces. Michigan Math. J. 62(3):657–664 [MR 3102534].
   Web of Science ®Google Scholar
• Calabri, A. (2002). On rational and ruled double planes. Ann. Mat. Pura Appl. (4) 181(4):365–387 [MR 1939687 (2003j:14049)].
   Web of Science ®Google Scholar
• Chase, S. U., Harrison, D. K., Rosenberg, A. (1965). Galois theory and Galois cohomology of commutative rings. Mem. Am. Math. Soc. 52:15–33 [MR 0195922 (33 #4118)].
   Google Scholar
• Childs, L. N. (1976). On Brauer groups of some normal local rings. In: Brauer groups (Proc. Conf., Northwestern Univ., Evanston, Ill., 1975). Lecture Notes in Mathematics, Vol. 549. Berlin: Springer, pp. 1–15 [MR 0432620 (55 #5607)].
   Google Scholar
• Colliot-Thélène, J.-L., Wittenberg, O. (2012). Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines. Am. J. Math. 134(5):1303–1327 [MR 2975237].
   Google Scholar
• Creutz, B., Viray, B. (2015). On Brauer groups of double covers of ruled surfaces. Math. Ann. 362(3–4):1169–1200 [MR 3368096].
   Google Scholar
• DeMeyer, F. R., Ford, T. J. (1992). Nontrivial, locally trivial Azumaya algebras. In: Azumaya Algebras, Actions, and Modules (Bloomington, IN, 1990). Contemporary Mathematics, Vol. 124. Providence, RI: American Mathematical Society, pp. 39–49 [MR 1144027 (93a:13001)].
   Google Scholar
• DeMeyer, F., Ford, T., Miranda, R. (1993). Rational singularities and the Brauer group. J. Algebra 162(2):287–294 [MR 94m:14042].
   Web of Science ®Google Scholar
• DeMeyer, F. R., Ingraham, E. (1971). Separable Algebras over Commutative Rings. Lecture Notes in Mathematics, Vol. 181. Berlin: Springer-Verlag [MR 0280479 (43 #6199)].
   Google Scholar
• Durfee, A. H. (1979). Fifteen characterizations of rational double points and simple critical points. Enseign. Math. (2) 25(1–2):131–163 [MR 543555 (80m:14003)].
   Google Scholar
• Ford, T. J. (1989). On the Brauer group of k[x1,…,xn,1∕f]. J. Algebra 122(2):410–424 [MR 999083 (90d:13005)].
   Web of Science ®Google Scholar
• Ford, T. J. (1992). On the Brauer group of a desingularization of a normal surface. Commun. Algebra 20(12): 3785–3791 [MR 1191981 (93j:13005)].
   Web of Science ®Google Scholar
• Ford, T. J. (1992). On the Brauer group of a localization. J. Algebra 147(2):365–378 [MR 1161299 (93c:13004)].
   Web of Science ®Google Scholar
• Ford, T. J. (2001). The Brauer group of an affine cone. J. Pure Appl. Algebra 155(1):29–40 [MR 1804326 (2001k:14044)].
   Web of Science ®Google Scholar
• Ford, T. J. (2013). The relative Brauer group of an affine double plane. Commun. Algebra 41(9):3277–3298 [MR 3169456].
   Web of Science ®Google Scholar
• Ford, T. J. (2014). The group of units on an affine variety. J. Algebra Appl. 13(8): 1450065 (27 pages, MR 3225132).
   Web of Science ®Google Scholar
• Ford, T. J. (2016). The relative Brauer group and generalized cyclic crossed products for a ramified covering. J. Algebra 450:1–58 [MR 3449685].
   Web of Science ®Google Scholar
• Fossum, R. M. (1973). The Divisor Class Group of a Krull Domain. New York: Springer-Verlag [MR 0382254 (52 #3139)].
   Google Scholar
• Göhner, H. (1975). Semifactoriality and Muhly’s condition (N) in two dimensional local rings. J. Algebra 34:403–429 [MR 0379489 (52 #394)].
   Web of Science ®Google Scholar
• Grothendieck, A. (1968). Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses. Dix Exposés sur la Cohomologie des Schémas. Amsterdam: North-Holland, pp. 46–66 [MR 0244269 (39 #5586a)].
   Google Scholar
• Grothendieck, A. (1968). Le groupe de Brauer. II. Théorie cohomologique. Dix Exposés sur la Cohomologie des Schémas. Amsterdam: North-Holland, pp. 67–87 [MR 0244270 (39 #5586b)].
   Google Scholar
• Grothendieck, A. (1968). Le groupe de Brauer. III. Exemples et compléments. Dix Exposés sur la Cohomologie des Schémas. Amsterdam: North-Holland, pp. 88–188 [MR 0244271 (39 #5586c)].
   Google Scholar
• Hartshorne, R. (1977). Algebraic Geometry. New York: Springer-Verlag [MR 0463157 (57 #3116)].
   Google Scholar
• Hirzebruch, F. (1995). The Topology of Normal Singularities of an Algebraic Surface (after D. Mumford). Séminaire Bourbaki, Vol. 8. Paris: Société Mathématique de France, Exp. No. 250, pp. 129–137 (MR 1611536).
   Google Scholar
• Hoobler, R. T. (1982). When is Br(X) = Br′(X)? In: van Oystaeyen, F., Verschoren, A., eds. Brauer Groups in Ring Theory and Algebraic Geometry (Wilrijk, 1981). Lecture Notes in Mathematics, Vol. 917. Berlin–New York: Springer, pp. 231–244 [MR 657433 (83g:14006)].
   Google Scholar
• Hoobler, R. T. (1984). Functors of graded rings. In: Van Oystaeyen, F., ed. Methods in Ring Theory (Antwerp, 1983). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 129. Dordrecht: Reidel, pp. 161–170 [MR 770588 (86f:16008)].
   Google Scholar
• Ieronymou, E., Skorobogatov, A. N. (2015). Odd order Brauer-Manin obstruction on diagonal quartic surfaces. Adv. Math. 270:181–205 [MR 3286534].
   Web of Science ®Google Scholar
• Ingalls, C., Obus, A., Ozman, E., Viray, B. Unramified Brauer classes on cyclic covers of the projective plane (preprint). arXiv:1310.8005 [math.AG].
   Google Scholar
• Kanzaki, T. (1968). On generalized crossed product and Brauer group. Osaka J. Math. 5:175–188 [MR 0241469 (39 #2809)].
   Google Scholar
• Lipman, J. (1969). Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36:195–279 [MR 0276239 (43 #1986)].
   Google Scholar
• Manin, Y. I. (1971). Le groupe de Brauer-Grothendieck en géométrie diophantienne. In: Actes du Congrès International des Mathématiciens (Nice, 1970). Tome 1. Paris: Gauthier-Villars, pp. 401–411 [MR 0427322 (55 #356)].
   Google Scholar
• Milne, J. S. (1980). Étale Cohomology. Princeton, N.J.: Princeton University Press [MR 559531 (81j:14002)].
   Google Scholar
• Mumford, D. (1961). The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. 9:5–22 [MR 0153682 (27 #3643)].
   Google Scholar
• Mumford, D. (1970). Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, Vol. 5. Bombay: Published for the Tata Institute of Fundamental Research [MR 0282985 (44 #219)].
   Google Scholar
• Orlik, P., Wagreich, P. (1971). Isolated singularities of algebraic surfaces with C* action. Ann. Math. (2) 93:205–228 [MR 0284435 (44 #1662)].
   Web of Science ®Google Scholar
• Rim, D. S. (1965). An exact sequence in Galois cohomology. Proc. Am. Math. Soc. 16:837–840 [MR 0179232 (31 #3480)].
   Web of Science ®Google Scholar
• Rotman, J. J. (1979). An Introduction to Homological Algebra. Pure and Applied Mathematics, Vol. 85. New York: Academic Press Inc. (Harcourt Brace Jovanovich Publishers) [MR 538169 (80k:18001)].
   Google Scholar
• Saltman, D. J. (1999). Lectures on Division Algebras. CBMS Regional Conference Series in Mathematics, Vol. 94. Providence, RI: American Mathematical Society [MR 1692654 (2000f:16023)].
   Google Scholar
• Scheja, G., Storch, U. (1984). Zur Konstruktion faktorieller graduierter Integritätsbereiche. Arch. Math. (Basel) 42(1):45–52 [MR 751470 (85m:13016)].
   Google Scholar
• Singh, A. K., Spiroff, S. (2007). Divisor class groups of graded hypersurfaces. In: Algebra, Geometry and Their Interactions. Contemporary Mathematics, Vol. 448. Providence, RI: American Mathematical Society, pp. 237–243 [MR 2389245 (2009f:13018)].
   Google Scholar
• van Geemen, B. (2005). Some remarks on Brauer groups of K3 surfaces. Adv. Math. 197(1):222–247 [MR 2166182 (2006h:14025)].
   Web of Science ®Google Scholar
• Watanabe, K. (1981). Some remarks concerning Demazure’s construction of normal graded rings. Nagoya Math. J. 83:203–211 [MR 632654 (83g:13016)].
   Web of Science ®Google Scholar