{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:11:55Z","timestamp":1776784315929,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"251","license":[{"start":{"date-parts":[[2006,1,27]],"date-time":"2006-01-27T00:00:00Z","timestamp":1138320000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of relations for generators of the ideal to compute its tight closure. In particular, our method gives an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis x Superscript a Baseline comma y Superscript a Baseline comma z Superscript a Baseline right-parenthesis Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>z<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(x^{a},y^{a},z^{a})^*<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K left-bracket x comma y comma z right-bracket slash left-parenthesis upper F right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">K[x,y,z]\/(F)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a homogeneous polynomial.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01730-8","type":"journal-article","created":{"date-parts":[[2005,4,12]],"date-time":"2005-04-12T12:20:25Z","timestamp":1113308425000},"page":"1495-1518","source":"Crossref","is-referenced-by-count":8,"title":["Computing the tight closure in dimension two"],"prefix":"10.1090","volume":"74","author":[{"given":"Holger","family":"Brenner","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,1,27]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/S0021-8693(03)00222-9","article-title":"Tight closure and projective bundles","volume":"265","author":"Brenner, Holger","year":"2003","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1090\/S0002-9947-03-03391-9","article-title":"Slopes of vector bundles on projective curves and applications to tight closure problems","volume":"356","author":"Brenner, Holger","year":"2004","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1017\/S0027763000014380","article-title":"\ud835\udc5d-ample bundles and their Chern classes","volume":"43","author":"Gieseker, David","year":"1971","journal-title":"Nagoya Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0027-7630","issn-type":"print"},{"issue":"4","key":"4","first-page":"228","article-title":"\u00c9l\u00e9ments de g\u00e9om\u00e9trie alg\u00e9brique. I. Le langage des sch\u00e9mas","author":"Grothendieck, A.","year":"1960","journal-title":"Inst. Hautes \\'{E}tudes Sci. Publ. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0073-8301","issn-type":"print"},{"issue":"5","key":"5","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1353\/ajm.1998.0037","article-title":"A characterization of rational singularities in terms of injectivity of Frobenius maps","volume":"120","author":"Hara, Nobuo","year":"1998","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1007\/BF01357141","article-title":"On the cohomology groups of moduli spaces of vector bundles on curves","volume":"212","author":"Harder, G.","year":"1974","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"29","key":"7","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BF02684806","article-title":"Ample vector bundles","author":"Hartshorne, Robin","year":"1966","journal-title":"Inst. Hautes \\'{E}tudes Sci. Publ. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0073-8301","issn-type":"print"},{"key":"8","series-title":"Graduate Texts in Mathematics, No. 52","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3849-0","volume-title":"Algebraic geometry","author":"Hartshorne, Robin","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902449"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1090\/conm\/159\/01508","article-title":"Solid closure","author":"Hochster, Melvin","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821851888"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"31","DOI":"10.2307\/1990984","article-title":"Tight closure, invariant theory, and the Brian\u00e7on-Skoda theorem","volume":"3","author":"Hochster, Melvin","year":"1990","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"key":"11","series-title":"CBMS Regional Conference Series in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1016\/0167-4889(95)00136-0","volume-title":"Tight closure and its applications","volume":"88","author":"Huneke, Craig","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/082180412X"},{"key":"12","isbn-type":"print","first-page":"187","article-title":"Tight closure, parameter ideals, and geometry","author":"Huneke, Craig","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/376435951X"},{"key":"13","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1515\/crll.1997.484.127","article-title":"Tight closure and the Kodaira vanishing theorem","volume":"484","author":"Huneke, Craig","year":"1997","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"14","series-title":"Aspects of Mathematics, E31","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-663-11624-0","volume-title":"The geometry of moduli spaces of sheaves","author":"Huybrechts, Daniel","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/3528069074"},{"key":"15","unstructured":"R. Lazarsfeld, Positivity in Algebraic Geometry (Preliminary Draft), 2001."},{"key":"16","unstructured":"G. J. Leuschke, Appendix: Some examples in tight closure, Trends in Commutative Algebra, MSRI publications, 51, 2004."},{"key":"17","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/BF01759316","article-title":"Some observations on cohomologically \ud835\udc5d-ample bundles","volume":"164","author":"Migliorini, Luca","year":"1993","journal-title":"Ann. Mat. Pura Appl. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4622","issn-type":"print"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"449","DOI":"10.2969\/aspm\/01010449","article-title":"The Chern classes and Kodaira dimension of a minimal variety","author":"Miyaoka, Yoichi","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0444703136"},{"key":"19","unstructured":"M. Katzman, Gr\u00f6bner bases and the Frobenius map, preprint, http:\/\/www.sheffield. ac.uk\/katzman\/Articles\/Publications.html."},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1006\/jabr.1997.7350","article-title":"A computation of tight closure in diagonal hypersurfaces","volume":"203","author":"Singh, Anurag K.","year":"1998","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"issue":"1","key":"21","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1215\/kjm\/1250518397","article-title":"Tight closure in graded rings","volume":"37","author":"Smith, Karen E.","year":"1997","journal-title":"J. Math. Kyoto Univ.","ISSN":"https:\/\/id.crossref.org\/issn\/0023-608X","issn-type":"print"},{"key":"22","isbn-type":"print","first-page":"353","article-title":"An introduction to tight closure","author":"Smith, Karen E.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/082470567X"},{"key":"23","unstructured":"S. Sullivant, Tight closure of monomial ideals in Fermat rings, preprint, 2002."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2005-74-251\/S0025-5718-05-01730-8\/S0025-5718-05-01730-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-251\/S0025-5718-05-01730-8\/S0025-5718-05-01730-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:11:20Z","timestamp":1776780680000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2005-74-251\/S0025-5718-05-01730-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1,27]]},"references-count":23,"journal-issue":{"issue":"251","published-print":{"date-parts":[[2005,7]]}},"alternative-id":["S0025-5718-05-01730-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01730-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2005,1,27]]}}}