{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T16:38:48Z","timestamp":1773247128850,"version":"3.50.1"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"287","license":[{"start":{"date-parts":[[2014,8,14]],"date-time":"2014-08-14T00:00:00Z","timestamp":1407974400000},"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>Let <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 2 comma 1 right-parenthesis right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>U<\/mml:mi>\n      <mml:mo>\u2286<\/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>\u00d7<\/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:mn>2<\/mml:mn>\n      <mml:mo>,<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\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}}(2,1))<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> be a basepoint free four-dimensional vector space. The sections corresponding to <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> determine a regular map <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>\u03d5<\/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>\u00d7<\/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\">\u27f6<\/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>. We study the associated bigraded ideal <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>\u2286<\/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> from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for <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>\u03d5<\/mml:mi>\n        <mml:mi>U<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo stretchy=\"false\">(<\/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>\u00d7<\/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\">)<\/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>, via work of Bus\u00e9-Jouanolou, Bus\u00e9-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper Z\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">Z<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathcal {Z}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. In four of the six cases <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I Subscript upper U\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>I<\/mml:mi>\n      <mml:mi>U<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">I_U<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular, this allows us to explicitly describe the implicit equation and singular locus of the image.<\/p>","DOI":"10.1090\/s0025-5718-2013-02764-0","type":"journal-article","created":{"date-parts":[[2013,8,14]],"date-time":"2013-08-14T15:48:48Z","timestamp":1376495328000},"page":"1337-1372","source":"Crossref","is-referenced-by-count":9,"title":["Syzygies and singularities of tensor product surfaces of bidegree (2,1)"],"prefix":"10.1090","volume":"83","author":[{"given":"Hal","family":"Schenck","sequence":"first","affiliation":[]},{"given":"Alexandra","family":"Seceleanu","sequence":"additional","affiliation":[]},{"given":"Javid","family":"Validashti","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,8,14]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/S0022-4049(99)00100-0","article-title":"Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions","volume":"150","author":"Aramova, Annetta","year":"2000","journal-title":"J. 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