{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:47:12Z","timestamp":1773251232718,"version":"3.50.1"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"9","license":[{"start":{"date-parts":[[2012,1,28]],"date-time":"2012-01-28T00:00:00Z","timestamp":1327708800000},"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":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    We show that given integers\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N greater-than-or-equal-to 2 comma left-parenthesis upper N comma d comma n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mo>\n                                \u2265\n                                \n                              <\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{N\\ge 2}, (N,d,n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"not-equals left-parenthesis 2 comma 2 comma 5 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2260\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ne (2,2,5)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N plus 1 less-than-or-equal-to n less-than-or-equal-to StartBinomialOrMatrix d plus upper N Choose upper N EndBinomialOrMatrix\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                              <mml:mrow>\n                                <mml:mstyle scriptlevel=\"0\">\n                                  <mml:mrow class=\"MJX-TeXAtom-OPEN\">\n                                    <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(<\/mml:mo>\n                                  <\/mml:mrow>\n                                <\/mml:mstyle>\n                                <mml:mfrac linethickness=\"0\">\n                                  <mml:mrow>\n                                    <mml:mi>d<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mi>N<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mi>N<\/mml:mi>\n                                <\/mml:mfrac>\n                                <mml:mstyle scriptlevel=\"0\">\n                                  <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\n                                    <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)<\/mml:mo>\n                                  <\/mml:mrow>\n                                <\/mml:mstyle>\n                              <\/mml:mrow>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{N+1\\le n\\le \\tbinom {d+N}{N}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , there is a family of\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    monomials 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 upper X 0 comma ellipsis comma upper X Subscript upper N Baseline right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mo>\n                                \u2026\n                                \n                              <\/mml:mo>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mi>N<\/mml:mi>\n                              <\/mml:msub>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">K\\left [X_0,\\ldots ,X_N\\right ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of degree\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that their syzygy bundle is stable. Case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N greater-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{N\\ge 3}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    was obtained independently by Coand\u01ce with a different choice of families of monomials.\n                  <\/p>\n                  <p>\n                    For\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis upper N comma d comma n right-parenthesis equals left-parenthesis 2 comma 2 comma 5 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{(N,d,n)=(2,2,5)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , there are\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5\">\n                        <mml:semantics>\n                          <mml:mn>5<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    monomials of degree\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2\">\n                        <mml:semantics>\n                          <mml:mn>2<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">2<\/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 upper X 0 comma upper X 1 comma upper X 2 right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">K\\left [X_0,X_1,X_2\\right ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that their syzygy bundle is semistable.\n                  <\/p>","DOI":"10.1090\/s0002-9939-2011-10745-7","type":"journal-article","created":{"date-parts":[[2011,1,28]],"date-time":"2011-01-28T16:36:19Z","timestamp":1296232579000},"page":"3155-3170","source":"Crossref","is-referenced-by-count":9,"special_numbering":"627","title":["Stability of syzygy bundles"],"prefix":"10.1090","volume":"139","author":[{"given":"Pedro","family":"Macias Marques","sequence":"first","affiliation":[]},{"given":"Rosa","family":"Mir\u00f3-Roig","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2011,1,28]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF02844672","article-title":"On the stability of certain higher rank bundles on \ud835\udc0f\u207f","volume":"41","author":"Ballico, Edoardo","year":"1992","journal-title":"Rend. Circ. Mat. Palermo (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0009-725X","issn-type":"print"},{"key":"2","unstructured":"[Bre] Holger Brenner, Semistable monomial syzygy bundles, personal notes."},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1016\/j.aim.2008.04.009","article-title":"Looking out for stable syzygy bundles","volume":"219","author":"Brenner, Holger","year":"2008","journal-title":"Adv. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/ulect\/042\/01","article-title":"Tight closure and vector bundles","author":"Brenner, Holger","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821844342"},{"issue":"7","key":"5","doi-asserted-by":"publisher","first-page":"1241","DOI":"10.1016\/j.jpaa.2009.10.009","article-title":"Stability and unobstructedness of syzygy bundles","volume":"214","author":"Costa, Laura","year":"2010","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"key":"6","unstructured":"[Coa09] Iustin Coand\u01ce, On the stability of syzygy bundles, arXiv:0909.4455, 2009."},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/BF02566370","article-title":"Restrictions of semistable bundles on projective varieties","volume":"59","author":"Flenner, Hubert","year":"1984","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"key":"8","unstructured":"[Mar09] Pedro Macias Marques, Stability and moduli spaces of syzygy bundles, Ph.D. thesis, Universitat de Barcelona, 2009."}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/2011-139-09\/S0002-9939-2011-10745-7\/S0002-9939-2011-10745-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/2011-139-09\/S0002-9939-2011-10745-7\/S0002-9939-2011-10745-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T22:11:25Z","timestamp":1773180685000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/2011-139-09\/S0002-9939-2011-10745-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,28]]},"references-count":8,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2011,9]]}},"alternative-id":["S0002-9939-2011-10745-7"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-2011-10745-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[2011,1,28]]}}}