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Journal articles on the topic 'Brauer groups of schemes'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 February 2022

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1

Biswas, Indranil, Ajneet Dhillon, and Jacques Hurtubise. "Brauer groups of Quot schemes." Michigan Mathematical Journal 64, no. 3 (September 2015): 493–508. http://dx.doi.org/10.1307/mmj/1441116655.

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2

Blass, Piotr, and Raymond Hoobler. "Picard and Brauer groups of Zariski schemes." Proceedings of the American Mathematical Society 97, no. 3 (March 1, 1986): 379. http://dx.doi.org/10.1090/s0002-9939-1986-0840613-9.

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3

Lieblich, Max. "Twisted sheaves and the period-index problem." Compositio Mathematica 144, no. 1 (January 2008): 1–31. http://dx.doi.org/10.1112/s0010437x07003144.

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AbstractWe use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong’s period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over algebraically closed fields to characteristic zero, which (among other things) extends de Jong’s result to include classes of period divisible by the characteristic of the base field. Finally, we use the theory developed here to give counterexamples to a standard type of local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.

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4

Lee, Heisook, and Morris Orzech. "Brauer groups and Galois cohomology for a Krull scheme." Journal of Algebra 95, no. 2 (August 1985): 309–31. http://dx.doi.org/10.1016/0021-8693(85)90106-1.

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5

Iyer, Jaya N. N., and Roy Joshua. "Brauer groups of schemes associated to symmetric powers of smooth projective curves in arbitrary characteristics." Journal of Pure and Applied Algebra 224, no. 3 (March 2020): 1009–22. http://dx.doi.org/10.1016/j.jpaa.2019.06.019.

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6

Colliot-Thélène, Jean-Louis, and Fei Xu. "Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms." Compositio Mathematica 145, no. 2 (March 2009): 309–63. http://dx.doi.org/10.1112/s0010437x0800376x.

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AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

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7

Antieau, Benjamin. "Cohomological obstruction theory for Brauer classes and the period-index problem." Journal of K-theory 8, no. 3 (December 13, 2010): 419–35. http://dx.doi.org/10.1017/is010011030jkt136.

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AbstractLet U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

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8

Le Bruyn, Lieven, and George Seelinger. "Fibers of Generic Brauer–Severi Schemes." Journal of Algebra 214, no. 1 (April 1999): 222–34. http://dx.doi.org/10.1006/jabr.1998.7656.

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9

Ma, Qixiao. "Specializing Brauer classes in Picard schemes." Journal of Pure and Applied Algebra 226, no. 2 (February 2022): 106832. http://dx.doi.org/10.1016/j.jpaa.2021.106832.

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10

Turull, Alexandre. "Characters, Brauer characters, and local Brauer groups." Communications in Algebra 49, no. 1 (July 29, 2020): 85–98. http://dx.doi.org/10.1080/00927872.2020.1793992.

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11

Bernardara, Marcello. "A semiorthogonal decomposition for Brauer-Severi schemes." Mathematische Nachrichten 282, no. 10 (October 2009): 1406–13. http://dx.doi.org/10.1002/mana.200610826.

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12

Seelinger, George F. "Brauer-Severi schemes of finitely generated algebras." Israel Journal of Mathematics 111, no. 1 (December 1999): 321–37. http://dx.doi.org/10.1007/bf02810690.

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13

Bertuccioni, Inta. "Brauer groups and cohomology." Archiv der Mathematik 84, no. 5 (May 2005): 406–11. http://dx.doi.org/10.1007/s00013-004-1202-0.

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14

Popescu, Cristian D., Jack Sonn, and Adrian R. Wadsworth. "n-Torsion of Brauer groups as relative Brauer groups of abelian extensions." Journal of Number Theory 125, no. 1 (July 2007): 26–38. http://dx.doi.org/10.1016/j.jnt.2006.09.012.

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15

González-Avilés, Cristian D. "Brauer groups and Néron class groups." International Journal of Number Theory 16, no. 10 (September 14, 2020): 2275–92. http://dx.doi.org/10.1142/s179304212050116x.

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Let [Formula: see text] be a global field and let [Formula: see text] be a finite set of primes of [Formula: see text] containing the Archimedean primes. We generalize the duality theorem for the Néron [Formula: see text]-class group of an abelian variety [Formula: see text] over [Formula: see text] established previously by removing the requirement that the Tate–Shafarevich group of [Formula: see text] be finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve [Formula: see text] over [Formula: see text] to a certain finite subquotient of the Brauer group of [Formula: see text].

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16

Chi, Wen-Chen, and Anly Li. "Groups of continuous characters and brauer groups." Communications in Algebra 26, no. 12 (January 1998): 4171–77. http://dx.doi.org/10.1080/00927879808826402.

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17

Moravec, Primož. "Unramified Brauer groups and isoclinism." Ars Mathematica Contemporanea 7, no. 2 (June 1, 2013): 337–40. http://dx.doi.org/10.26493/1855-3974.392.9fd.

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18

Bartel, Alex, and Tim Dokchitser. "Brauer relations in finite groups." Journal of the European Mathematical Society 17, no. 10 (2015): 2473–512. http://dx.doi.org/10.4171/jems/563.

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19

Edidin, Dan, Brendan Hassett, Andrew Kresch, and Angelo Vistoli. "Brauer groups and quotient stacks." American Journal of Mathematics 123, no. 4 (2001): 761–77. http://dx.doi.org/10.1353/ajm.2001.0024.

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20

Chen, Huixiang, and Yinhuo Zhang. "Cocycle Deformations and Brauer Groups." Communications in Algebra 35, no. 2 (January 26, 2007): 399–433. http://dx.doi.org/10.1080/00927870601052422.

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21

NENCIU, ADRIANA. "BRAUER PAIRS OF VZ-GROUPS." Journal of Algebra and Its Applications 07, no. 05 (October 2008): 663–70. http://dx.doi.org/10.1142/s0219498808003065.

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Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserve all the character values and the power map. A group is called a VZ-group if all its nonlinear irreducible characters vanish off the center. In this paper we give necessary and sufficient conditions for two non-isomorphic VZ-groups to form a Brauer pair.

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22

Baranovsky, Vladimir, and Tihomir Petrov. "Brauer groups and crepant resolutions." Advances in Mathematics 209, no. 2 (March 2007): 547–60. http://dx.doi.org/10.1016/j.aim.2006.05.009.

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23

Fong, Paul, and Bhama Srinivasan. "Brauer trees in classical groups." Journal of Algebra 131, no. 1 (May 1990): 179–225. http://dx.doi.org/10.1016/0021-8693(90)90172-k.

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24

MacQuarrie, John, and Peter Symonds. "Brauer theory for profinite groups." Journal of Algebra 398 (January 2014): 496–508. http://dx.doi.org/10.1016/j.jalgebra.2013.09.004.

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25

Cegarra, A. M., and A. R. Garzón. "Equivariant Brauer groups and cohomology." Journal of Algebra 296, no. 1 (February 2006): 56–74. http://dx.doi.org/10.1016/j.jalgebra.2005.11.032.

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26

Bertolin, Cristiana, and Federica Galluzzi. "Brauer groups of 1-motives." Journal of Pure and Applied Algebra 225, no. 11 (November 2021): 106754. http://dx.doi.org/10.1016/j.jpaa.2021.106754.

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27

Moravec, Primož. "Unramified Brauer groups of finite and infinite groups." American Journal of Mathematics 134, no. 6 (2012): 1679–704. http://dx.doi.org/10.1353/ajm.2012.0046.

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28

Hormozi, Mahdi, and Kijti Rodtes. "Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters." Electronic Journal of Linear Algebra 31 (February 5, 2016): 263–85. http://dx.doi.org/10.13001/1081-3810.3155.

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This paper provides some properties of Brauer symmetry classes of tensors. A dimension formula is derived for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups whose non-linear Brauer characters have support being a cyclic group. Using the derived formula, necessary and sufficient condition are investigated for the existence of an o-basis of dicyclic groups, semi-dihedral groups, and also those things are reinvestigated on dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.

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29

Skorobogatov, Alexei N., and Yuri G. Zarhin. "Kummer varieties and their Brauer groups." Pure and Applied Mathematics Quarterly 13, no. 2 (2017): 337–68. http://dx.doi.org/10.4310/pamq.2017.v13.n2.a5.

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30

Efrat, Ido. "On fields with finite Brauer groups." Pacific Journal of Mathematics 177, no. 1 (January 1, 1997): 33–46. http://dx.doi.org/10.2140/pjm.1997.177.33.

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31

Sadhu, Vivek. "Relative Brauer groups and étale cohomology." Journal of Pure and Applied Algebra 224, no. 12 (December 2020): 106428. http://dx.doi.org/10.1016/j.jpaa.2020.106428.

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32

Bright, Martin. "Brauer groups of diagonal quartic surfaces." Journal of Symbolic Computation 41, no. 5 (May 2006): 544–58. http://dx.doi.org/10.1016/j.jsc.2005.10.001.

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33

Baker, Andrew, Birgit Richter, and Markus Szymik. "Brauer groups for commutative S-algebras." Journal of Pure and Applied Algebra 216, no. 11 (November 2012): 2361–76. http://dx.doi.org/10.1016/j.jpaa.2012.03.001.

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34

Hoshi, Akinari, Ming-Chang Kang, and Boris E. Kunyavskii. "Noether’s problem and unramified Brauer groups." Asian Journal of Mathematics 17, no. 4 (2013): 689–714. http://dx.doi.org/10.4310/ajm.2013.v17.n4.a8.

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35

Aljadeff, Eli, and Jack Sonn. "Relative Brauer groups and $m$-torsion." Proceedings of the American Mathematical Society 130, no. 5 (November 9, 2001): 1333–37. http://dx.doi.org/10.1090/s0002-9939-01-06286-4.

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36

Aravire, Roberto, and Bill Jacob. "Relative Brauer groups in characteristic $p$." Proceedings of the American Mathematical Society 137, no. 04 (November 13, 2008): 1265–73. http://dx.doi.org/10.1090/s0002-9939-08-09746-3.

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37

Hiss, Gerhard. "Groups whose Brauer-characters are liftable." Journal of Algebra 94, no. 2 (June 1985): 388–405. http://dx.doi.org/10.1016/0021-8693(85)90192-9.

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38

Diaz, H. Anthony. "Galois descent for higher Brauer groups." manuscripta mathematica 163, no. 3-4 (November 26, 2019): 537–51. http://dx.doi.org/10.1007/s00229-019-01170-5.

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39

Fein, Burton, and Murray Schacher. "Brauer groups of algebraic function fields." Journal of Algebra 103, no. 2 (October 1986): 454–65. http://dx.doi.org/10.1016/0021-8693(86)90146-8.

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40

Eick, Bettina, and Jürgen Müller. "On p-groups forming Brauer pairs." Journal of Algebra 304, no. 1 (October 2006): 286–303. http://dx.doi.org/10.1016/j.jalgebra.2005.08.021.

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41

Navarro, Gabriel, Lucia Sanus, and Pham Huu Tiep. "Groups with two real Brauer characters." Journal of Algebra 307, no. 2 (January 2007): 891–98. http://dx.doi.org/10.1016/j.jalgebra.2005.12.001.

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42

Jahn, Thomas. "The order of higher Brauer groups." Mathematische Annalen 362, no. 1-2 (September 30, 2014): 43–54. http://dx.doi.org/10.1007/s00208-014-1105-8.

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43

Kresch, Andrew, and Yuri Tschinkel. "Brauer groups of involution surface bundles." Pure and Applied Mathematics Quarterly 17, no. 2 (2021): 649–69. http://dx.doi.org/10.4310/pamq.2021.v17.n2.a4.

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44

Sonn, Jack. "Brauer groups, embedding problems, and nilpotent groups as Galois groups." Israel Journal of Mathematics 85, no. 1-3 (February 1994): 391–405. http://dx.doi.org/10.1007/bf02758649.

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45

Bogomolov, F. A. "BRAUER GROUPS OF FIELDS OF INVARIANTS OF ALGEBRAIC GROUPS." Mathematics of the USSR-Sbornik 66, no. 1 (February 28, 1990): 285–99. http://dx.doi.org/10.1070/sm1990v066n01abeh001173.

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46

Moravec, Primož. "Groups of order p5 and their unramified Brauer groups." Journal of Algebra 372 (December 2012): 420–27. http://dx.doi.org/10.1016/j.jalgebra.2012.10.002.

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47

O’HARA, JAMES E., HIROSHI SHIMA, and CHUNTIAN ZHANG. "ANNOTATED CATALOGUE OF THE TACHINIDAE (INSECTA: DIPTERA) OF CHINA." Zootaxa 2190, no. 1 (August 6, 2009): 1–236. http://dx.doi.org/10.11646/zootaxa.2190.1.1.

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The Tachinidae of mainland China and Taiwan (generally referred to as China herein for brevity) are catalogued. A total of 1109 valid species are recorded of which 403 species (36%) are recorded as endemic. Distributions within China are given according to the 33 administrative divisions of the country, and distributions outside China are given according to a scheme of geographical divisions developed for this catalogue and most finely divided for the Palaearctic and Oriental Regions. The catalogue is based on examination of the primary literature comprising about 670 references and also includes a small number of records based on unpublished data from specimens examined in collections. Taxa are arranged hierarchically under the categories of subfamily, tribe, genus, subgenus (where recognized), and species. Nomenclatural details are provided for nominal genera and species. This includes synonyms at both levels for taxa described or recorded from China. For valid species, distributions are provided along with complete name-bearing type data for associated names. Additional information is given in the form of notes, numbering more than 300 in the catalogue section and about 50 in the references section. Six genera are newly recorded from China: Calliethilla Shima (Ethillini), Chetoptilia Rondani (Dufouriini), Demoticoides Mesnil (Leskiini), Pseudalsomyia Mesnil (Goniini), Redtenbacheria Schiner (Eutherini), and Rutilia Robineau-Desvoidy (Rutiliini). Fourteen species are newly recorded from China: Actia solida Tachi & Shima, Atylostoma towadensis (Matsumura), Chetoptilia burmanica (Baranov), Demoticoides pallidus Mesnil, Dexiosoma lineatum Mesnil, Feriola longicornis Mesnil, Frontina femorata Shima, Phebellia laxifrons Shima, Prodegeeria gracilis Shima, Prooppia stulta (Zetterstedt), Redtenbacheria insignis Egger, Sumpigaster subcompressa (Walker), Takanomyia frontalis Shima, and Takanomyia rava Shima. Two genera and 23 species are recorded as misidentified from China. New names are proposed for three preoccupied names: Pseudodexilla O’Hara, Shima & Zhang, nomen novum for Pseudodexia Chao, 2002; Admontia longicornalis O’Hara, Shima & Zhang, nomen novum for Admontia longicornis Yang & Chao, 1990; and Erythrocera neolongicornis O’Hara, Shima & Zhang, nomen novum for Pexopsis longicornis Sun & Chao, 1993. New type species fixations are made under the provisions of Article 70.3.2 of ICZN (1999) for 13 generic names: Chetoliga Rondani, Discochaeta Brauer & Bergenstamm, Erycina Mesnil, Eurigaster Macquart, Microvibrissina Villeneuve, Oodigaster Macquart, Plagiopsis Brauer & Bergenstamm, Prooppia Townsend, Ptilopsina Villeneuve, Ptilotachina Brauer & Bergenstamm, Rhinotachina Brauer & Bergenstamm, Schaumia Robineau-Desvoidy, and Setigena Brauer & Bergenstamm. Subgenus Tachina (Servillia Robineau-Desvoidy) is reduced to a synonym of subgenus Tachina (Tachina Meigen). The valid names of two species are reduced to nomina nuda and replaced by other available names with new status as valid names: Siphona (Aphantorhaphopsis) perispoliata (Mesnil) replaces S. (A.) mallochiana (Gardner), and Zenillia terrosa Mesnil replaces Z. grisellina (Gardner). The following 12 new combinations are proposed: Carcelina shangfangshanica (Chao & Liang), Drino (Drino) interfrons (Sun & Chao), Drino (Zygobothria) hirtmacula (Liang & Chao), Erythrocera longicornis (Sun & Chao) (a preoccupied name and replaced with Erythrocera neolongicornis O’Hara, Shima & Zhang, nomen novum), Isosturmia aureipollinosa (Chao & Zhou), Isosturmia setamacula (Chao & Liang), Isosturmia setula (Liang & Chao), Paratrixa flava (Shi), Phryno jilinensis (Sun), Phryno tibialis (Sun), Prosopodopsis ruficornis (Chao), and Takanomyia parafacialis (Sun & Chao). The following 19 new synonymies are proposed: Atylomyia chinensis Zhang & Ge with Tachina parallela Meigen (current name Bessa parallela), Atylomyia minutiungula Zhang & Wang with Ptychomyia remota Aldrich (current name Bessa remota), Carcelia (Carcelia) hainanensis Chao & Liang with Carcelia rasoides Baranov, Carcelia frontalis Baranov with Carcelia caudata Baranov, Carcelia hirtspila Chao & Shi with Carcelia (Parexorista) delicatula Mesnil (current name Carcelia (Euryclea) delicatula), Carcelia septima Baranov with Carcelia octava Baranov, Carcelia (Senometopia) dominantalis Chao & Liang with Carcelia quarta Baranov (current name Senometopia quarta), Carcelia (Senometopia) maculata Chao & Liang with Carcelia octava Baranov, Drino hersei Liang & Chao with Sturmia atropivora RobineauDesvoidy (current name Drino (Zygobothria) atropivora), Eucarcelia nudicauda Mesnil with Carcelia octava Baranov, Isopexopsis Sun & Chao with Takanomyia Mesnil, Mikia nigribasicosta Chao & Zhou withBombyliomyia apicalis Matsumura (current name Mikia apicalis), Parasetigena jilinensis Chao & Mao with Phorocera (Parasetigena) agilis takaoi Mesnil (current name Parasetigena takaoi), Phebellia latisurstyla Chao & Chen with Phebellia latipalpis Shima (current name Prooppia latipalpis), Servillia linabdomenalis Chao with Servillia cheni Chao (current name Tachina (Tachina) cheni), Servillia planiforceps Chao with Tachina sobria Walker, Spiniabdomina Shi with Paratrixa Brauer & Bergenstamm, Tachina kunmingensis Chao & Arnaud with Tachina sobria Walker, and Thecocarcelia tianpingensis Sun & Chao with Drino (Isosturmia) chatterjeeana japonica Mesnil (current name Isosturmia japonica). Musca libatrix Panzer is a nomen protectum and Musca libatrix Scopoli and Musca libatrix Geoffroy are nomina oblita. Similarly, Redtenbacheria insignis Egger is a nomen protectum and Redtenbacheria spectabilis Schiner is a nomen oblitum. Lectotypes are designated for the following 12 nominal species based on name-bearing type material in CNC: Akosempomyia caudata Villeneuve, Blepharipoda schineri Mesnil, Carcelia puberula Mesnil, Compsoptesis phoenix Villeneuve, Ectophasia antennata Villeneuve, Gymnosoma brevicorne Villeneuve, Kosempomyia tibialis Villeneuve, Phasia pusilla Meigen, Tachina fallax pseudofallax Villeneuve, Tachina chaoi Mesnil, Wagneria umbrinervis Villeneuve, and Zambesa claripalpis Villeneuve.China is an expansive country of 9.6 million square kilometers in eastern Asia. It is a land of physical and ecological extremes: southern subtropical and tropical forests, richly diverse southwestern mountains, towering Himalayas, harsh and inhospitable Tibetan Plateau, western Tien Shan range, dry Taklimakan and Goli Deserts, northeastern temperate broadleaf and coniferous forests, and eastern fertile plains and lesser mountains. Along its southern and western borders are portions of four of the world’s 34 “biodiversity hotspots”, places recognized by Conservation International for their high endemicity and threatened habitat. These are the Indo-Burma hotspot, Mountains of Southwest China hotspot (particularly Hengduan Shan), Himalaya hotspot, and Mountains of Central Asia hotspot (represented in China by Tien Shan) (http:// www.biodiversityhotspots.org). These biodiversity hotspots, and other biodiverse places in China, have given rise to an endemic fauna and flora of significant size. In the plant world, for example, the Hengduan Shan is known as the hotbed of Rhododendron evolution with about 230 species. Among the vertebrates are such Chinese endemics as the giant panda (Ailuropoda melanoleuca), golden monkeys (Rhinopithecus spp.), baiji (Lipotes vexillifer), and brown eared pheasant (Crossoptilon mantchuricum). Less conspicuous, but many times more numerous in species, are the endemic invertebrates that have evolved within present-day China. Biogeographically, China is unique among the countries of the world in lying at the crossroads of the Palaearctic and Oriental Regions. Hence, for most groups of organisms, the species of China consist of a combination of Palaearctic, Oriental, and endemic elements. This is true also of the Tachinidae of China. The Tachinidae are one of the largest families of Diptera with almost 10,000 described species and many thousands of undescribed species (Stireman et al. 2006). The family is correspondingly diverse in China, but because the Chinese tachinid fauna is still in a period of discovery and study, it must be significantly larger than the numbers given here might suggest. We record 1109 species and 257 genera of Tachinidae from mainland China and Taiwan, the former number representing about 11% of the world’s described tachinid species. From mainland China we record 1040 species, which compares to 754 and 832 species recorded from the same area by Chao et al. (1998) and Hua (2006), respectively. Our higher number is partly a reflection of species described from China since those works, or described from elsewhere and recently recognized from China, but a significant number of species were presumably overlooked by Chao et al. (1998) and Hua (2006) in the voluminous literature that exists on Chinese insects. The Chinese tachinid fauna has very few endemic genera and none of significant size, but has 403 species recorded as endemic to China plus Taiwan. This represents 36% of the total tachinid fauna. We record 343 species as endemic to mainland China and 32 species as endemic to Taiwan. The total number of species recorded from Taiwan is 231; some of these species are shared with the Oriental Region but not with mainland China.

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48

Efrat, Ido, and Eliyahu Matzri. "Vanishing of Massey Products and Brauer Groups." Canadian Mathematical Bulletin 58, no. 4 (December 1, 2015): 730–40. http://dx.doi.org/10.4153/cmb-2015-026-5.

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AbstractLet p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. We prove a stronger form of the vanishing property for global fields.

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Garbagnatti, Alice, and Matthias Schütt. "Enriques surfaces: Brauer groups and Kummer structures." Michigan Mathematical Journal 61, no. 2 (June 2012): 297–330. http://dx.doi.org/10.1307/mmj/1339011529.

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Bright, Martin. "Brauer groups of singular del Pezzo surfaces." Michigan Mathematical Journal 62, no. 3 (September 2013): 657–64. http://dx.doi.org/10.1307/mmj/1378757892.

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