{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T11:32:55Z","timestamp":1648812775441},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5032,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2000,6]]},"abstract":"<jats:p>Let <jats:italic>k \u2282 K<\/jats:italic> be a field extension, where <jats:italic>K<\/jats:italic> is an algebraically closed field of any characteristic and <jats:italic>k<\/jats:italic> is the prime field. Recall the following property of Hilbert Schemes (see, for example, [1], Proposition 1.16): Suppose <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline1\" \/> \u2282 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline2\" \/> \u00d7 <jats:italic>S<\/jats:italic> is a flat family of closed subschemes of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline2\" \/> parametrised by a scheme <jats:italic>S\/k<\/jats:italic>. Then for every closed subscheme <jats:italic>Z<\/jats:italic> \u2282 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline2\" \/> in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline1\" \/>, if [<jats:italic>Z<\/jats:italic>] denotes the Hilbert point of <jats:italic>Z<\/jats:italic> in Hilb(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline3\" \/>) then the residue field of Hilb(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline3\" \/>) at [<jats:italic>Z<\/jats:italic>] is the minimal field of definition for <jats:italic>Z<\/jats:italic>. Intuitively, this says that as a family parametrised by Hilb(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline3\" \/>), each fibre of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline1\" \/> lies above a point whose \u201cco-ordinates\u201d generate its minimal field of definition.<\/jats:p><jats:p>In the following note we are concerned with a more naive form of the above situation, for which we wish to give an elementary account. Suppose \u03d5<jats:italic>(x,y)<\/jats:italic> is a system of polynomial equations over <jats:italic>k<\/jats:italic> (in variables <jats:italic>x = (x<jats:sub>1<\/jats:sub>,\u2026, x<jats:sub>m<\/jats:sub>)<\/jats:italic> and parameters <jats:italic>y<\/jats:italic> = (<jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>y<jats:sub>n<\/jats:sub><\/jats:italic>)), such that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>is a family of (possibly reducible) affine varieties in <jats:italic>K<jats:sup>m<\/jats:sup><\/jats:italic>. Each nonempty member of this family has a unique minimal field of definition. The question arises as to whether it is possible to express this family of varieties using parameters that come, pointwise, from these minimal fields of definition. That is, is there a system of polynomial equations <jats:italic>\u03c8(x, z)<\/jats:italic> over <jats:italic>k<\/jats:italic>, such that each <jats:italic>\u03c8(x, b)<\/jats:italic> with <jats:italic>b<\/jats:italic> \u2208 <jats:italic>K<jats:sup>N<\/jats:sup><\/jats:italic> is of the form <jats:italic>V<jats:sub>a<\/jats:sub><\/jats:italic> for some <jats:italic>a<\/jats:italic> \u2208 <jats:italic>K<jats:sup>n<\/jats:sup><\/jats:italic>; and such that each <jats:italic>V<jats:sub>a<\/jats:sub><\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200012226_inline1\" \/> is defined by <jats:italic>\u03c8(x, b)<\/jats:italic> for some <jats:italic>b<\/jats:italic> \u2208 <jats:italic>K<jats:sup>N<\/jats:sup><\/jats:italic> whose coordinates generate the minimal field of definition for <jats:italic>V<jats:sub>a<\/jats:sub><\/jats:italic>? Moreover, we would like <jats:italic>b<\/jats:italic> to be obtained definably from <jats:italic>a<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2586572","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:04:37Z","timestamp":1146924277000},"page":"817-821","source":"Crossref","is-referenced-by-count":0,"title":["A note on uniform definability and minimal fields of definition"],"prefix":"10.1017","volume":"65","author":[{"given":"Rahim","family":"Moosa","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200012226_ref002","volume-title":"Introduction to algebraic geometry","author":"Lang","year":"1958"},{"key":"S0022481200012226_ref001","volume-title":"Rational curves on algebraic varieties","author":"Koll\u00e1r","year":"1991"},{"key":"S0022481200012226_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388493"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200012226","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T17:21:06Z","timestamp":1557336066000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200012226\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2000,6]]}},"alternative-id":["S0022481200012226"],"URL":"https:\/\/doi.org\/10.2307\/2586572","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,6]]}}}