{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:28:53Z","timestamp":1775467733137,"version":"3.50.1"},"reference-count":6,"publisher":"American Mathematical Society (AMS)","issue":"5","license":[{"start":{"date-parts":[[2007,11,13]],"date-time":"2007-11-13T00:00:00Z","timestamp":1194912000000},"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>In this note it is shown that, given a smooth minimal complex surface of general type <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n  <mml:semantics>\n    <mml:mi>S<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Subscript g Baseline left-parenthesis upper S right-parenthesis equals 0\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>p<\/mml:mi>\n        <mml:mi>g<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>S<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p_g(S)=0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript upper S Superscript 2 Baseline equals 3\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msubsup>\n        <mml:mi>K<\/mml:mi>\n        <mml:mi>S<\/mml:mi>\n        <mml:mn>2<\/mml:mn>\n      <\/mml:msubsup>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>3<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">K^2_S=3<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, for which the bicanonical map <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi Subscript 2 upper K\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>\u03c6<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mn>2<\/mml:mn>\n        <mml:mi>K<\/mml:mi>\n      <\/mml:mrow>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\varphi _{2K}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a morphism, the degree of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi Subscript 2 upper K\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>\u03c6<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mn>2<\/mml:mn>\n        <mml:mi>K<\/mml:mi>\n      <\/mml:mrow>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\varphi _{2K}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is not 3. This completes our earlier results, showing that if <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n  <mml:semantics>\n    <mml:mi>S<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a minimal surface of general type with <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Subscript g Baseline equals 0\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>p<\/mml:mi>\n        <mml:mi>g<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">p_g=0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K squared greater-than-or-equal-to 3\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>K<\/mml:mi>\n        <mml:mn>2<\/mml:mn>\n      <\/mml:msup>\n      <mml:mo>\u2265<\/mml:mo>\n      <mml:mn>3<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">K^2\\ge 3<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> such that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue 2 upper K Subscript upper S Baseline EndAbsoluteValue\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:msub>\n        <mml:mi>K<\/mml:mi>\n        <mml:mi>S<\/mml:mi>\n      <\/mml:msub>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">|2K_S|<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is free, then the bicanonical map of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n  <mml:semantics>\n    <mml:mi>S<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> can have degree 1, 2 or 4.<\/p>","DOI":"10.1090\/s0002-9939-06-08633-3","type":"journal-article","created":{"date-parts":[[2007,2,2]],"date-time":"2007-02-02T19:59:54Z","timestamp":1170446394000},"page":"1279-1282","source":"Crossref","is-referenced-by-count":7,"special_numbering":"575","title":["The degree of the bicanonical map of a surface with \ud835\udc5d_{\ud835\udc54}=0"],"prefix":"10.1090","volume":"135","author":[{"given":"Margarida","family":"Mendes Lopes","sequence":"first","affiliation":[]},{"given":"Rita","family":"Pardini","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"1","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-96754-2","volume-title":"Compact complex surfaces","volume":"4","author":"Barth, W.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/3540121722"},{"issue":"5","key":"2","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1007\/s000130050142","article-title":"The degree of the generators of the canonical ring of surfaces of general type with \ud835\udc5d_{\ud835\udc54}=0","volume":"69","author":"Mendes Lopes, Margarida","year":"1997","journal-title":"Arch. 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