{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T13:05:49Z","timestamp":1775480749835,"version":"3.50.1"},"reference-count":20,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":1127,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2007]]},"abstract":"<jats:p>We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of III [2], for families of curves of genus 2 of the form <jats:italic>y<\/jats:italic><jats:sup>2<\/jats:sup> = <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>), where <jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic>) is a quintic containing an irreducible cubic factor.<\/jats:p>","DOI":"10.1112\/s1461157000001455","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T11:42:31Z","timestamp":1375789351000},"page":"354-377","source":"Crossref","is-referenced-by-count":16,"title":["The Brauer-Manin Obstruction and III[2]"],"prefix":"10.1112","volume":"10","author":[{"given":"M.J.","family":"Bright","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N.","family":"Bruin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"E.V","family":"Flynn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Logan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000001455_ref020","unstructured":"20. Wittenberg O. , \u2018Principe de Hasse pour les surfaces de del Pezzo de degr\u00e9 4\u2019, Ph.D. thesis, Universit\u00e9 Paris XI Orsay, 2005."},{"key":"S1461157000001455_ref018","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100076106"},{"key":"S1461157000001455_ref017","doi-asserted-by":"publisher","DOI":"10.4064\/aa98-3-4"},{"key":"S1461157000001455_ref012","first-page":"401","article-title":"\u2018Le groupe de Brauer-Grothendieck en g\u00e9om\u00e9trie diophantienne\u2019","volume":"1","author":"Manin","year":"1970","journal-title":"Actes du Congr\u00e8s International des Math\u00e9maticiens"},{"key":"S1461157000001455_ref011","unstructured":"11. Logan A. , \u2018MAGMA programs for computing the Brauer-Manin obstruction on a del Pezzo surface of degree 4\u2019, http:\/\/www.math.uwaterloo.ca\/~a51ogan\/math\/index.html."},{"key":"S1461157000001455_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0092227"},{"key":"S1461157000001455_ref006","first-page":"72","article-title":"\u2018Intersections of two quadrics and Ch\u00e2telet surfaces. II\u2019","volume":"374","author":"Colliot-Th\u00e9l\u00e8ne","year":"1987","journal-title":"J. Reine Angew. Math"},{"key":"S1461157000001455_ref005","first-page":"37","article-title":"\u2018Intersections of two quadrics and Ch\u00e2telet surfaces. I\u2019","volume":"373","author":"Colliot-Th\u00e9l\u00e8ne","year":"1987","journal-title":"J. Reine Angew. Math"},{"key":"S1461157000001455_ref003","unstructured":"3. Bruin N. and Logan A. , \u2018Electronic resources\u2019, http:\/\/www.cecm.sfu.ca\/~nbruin\/BM0bstrAndSha."},{"key":"S1461157000001455_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004106009844"},{"key":"S1461157000001455_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/s002220050291"},{"key":"S1461157000001455_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0078705"},{"key":"S1461157000001455_ref010","article-title":"\u2018del Pezzo surfaces of degree four\u2019","volume":"113","author":"Kunyavski\u012d","year":"1989","journal-title":"M\u00e9m. Soc. Math. France"},{"key":"S1461157000001455_ref019","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004198003144"},{"key":"S1461157000001455_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2005.10.007"},{"key":"S1461157000001455_ref014","unstructured":"14. Reid M. , \u2018Intersections of two quadrics\u2019, PhD thesis, University of Cambridge, 1972."},{"key":"S1461157000001455_ref008","first-page":"46","volume-title":"Dix expos\u00e9s sur la cohomologie des sch\u00e9mas","author":"Grothendieck","year":"1968"},{"key":"S1461157000001455_ref007","doi-asserted-by":"publisher","DOI":"10.5802\/afst.751"},{"key":"S1461157000001455_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/s002080050156"},{"key":"S1461157000001455_ref013","unstructured":"13. Manin Yu.I. , Cubic forms: algebra, geometry, arithmetic, translated from the Russian by Hazewinkel M. , North-Holland Mathematical Library 4 (North-Holland, Amsterdam, 1974)."}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157000001455","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,6]],"date-time":"2019-06-06T22:01:14Z","timestamp":1559858474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157000001455\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007]]},"references-count":20,"alternative-id":["S1461157000001455"],"URL":"https:\/\/doi.org\/10.1112\/s1461157000001455","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007]]}}}