{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T21:26:23Z","timestamp":1773264383578,"version":"3.50.1"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"1","license":[{"start":{"date-parts":[[2016,6,9]],"date-time":"2016-06-09T00:00:00Z","timestamp":1465430400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 1312071"],"award-info":[{"award-number":["DMS 1312071"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper U subset-of-or-equal-to upper H Superscript 0 Baseline left-parenthesis script upper O Subscript double-struck upper P Sub Superscript 1 Subscript times double-struck upper P Sub Superscript 1 Subscript Baseline left-parenthesis a comma b right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>U<\/mml:mi>\n                            <mml:mo>\n                              \u2286\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msub>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:msup>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                                    <\/mml:mrow>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:msup>\n                                  <mml:mo>\n                                    \u00d7\n                                    \n                                  <\/mml:mo>\n                                  <mml:msup>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                                    <\/mml:mrow>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:msup>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">U \\subseteq H^0({\\mathcal {O}_{\\mathbb {P}^1 \\times \\mathbb {P}^1}}(a,b))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a basepoint free four-dimensional vector space, with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a comma b greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a,b \\ge 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The sections corresponding to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper U\">\n                        <mml:semantics>\n                          <mml:mi>U<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">U<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    determine a regular map\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi Subscript upper U Baseline colon double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1 Baseline long right-arrow double-struck upper P cubed\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03d5\n                                \n                              <\/mml:mi>\n                              <mml:mi>U<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msup>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msup>\n                              <mml:mo>\n                                \u00d7\n                                \n                              <\/mml:mo>\n                              <mml:msup>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">\n                              \u27f6\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi _U: {\\mathbb {P}^1 \\times \\mathbb {P}^1} \\longrightarrow \\mathbb {P}^3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We show that there can be at most one linear syzygy on the associated bigraded ideal\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I Subscript upper U Baseline subset-of-or-equal-to k left-bracket s comma t semicolon u comma v right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>I<\/mml:mi>\n                              <mml:mi>U<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2286\n                              \n                            <\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>v<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">I_U \\subseteq k[s,t;u,v]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Existence of a linear syzygy, coupled with the assumption that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper U\">\n                        <mml:semantics>\n                          <mml:mi>U<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">U<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is basepoint free, implies the existence of an additional \u201cspecial pair\u201d of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi Subscript upper U Baseline left-parenthesis double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03d5\n                                \n                              <\/mml:mi>\n                              <mml:mi>U<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi _U(\\mathbb {P}^1 \\times \\mathbb {P}^1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper S normal i normal n normal g left-parenthesis phi Subscript upper U Baseline left-parenthesis double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1 Baseline right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">S<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">n<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">g<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03d5\n                                \n                              <\/mml:mi>\n                              <mml:mi>U<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                 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