{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T09:26:31Z","timestamp":1775813191146,"version":"3.50.1"},"reference-count":44,"publisher":"American Mathematical Society (AMS)","issue":"4","license":[{"start":{"date-parts":[[2007,10,17]],"date-time":"2007-10-17T00:00:00Z","timestamp":1192579200000},"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":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K squared equals 1\">\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>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">K^2=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and are usually called\n                    <italic>numerical Godeaux surfaces<\/italic>\n                    . Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group.\n                  <\/p>","DOI":"10.1090\/s0002-9947-06-04110-9","type":"journal-article","created":{"date-parts":[[2006,12,19]],"date-time":"2006-12-19T08:04:15Z","timestamp":1166515455000},"page":"1605-1632","source":"Crossref","is-referenced-by-count":18,"special_numbering":"863","title":["Numerical Godeaux surfaces with an involution"],"prefix":"10.1090","volume":"359","author":[{"given":"Alberto","family":"Calabri","sequence":"first","affiliation":[]},{"given":"Ciro","family":"Ciliberto","sequence":"additional","affiliation":[]},{"given":"Margarida","family":"Mendes Lopes","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,10,17]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"546","DOI":"10.2307\/1970717","article-title":"The index of elliptic operators. 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Borrelli, On the classification of surfaces of general type with non-birational bicanonical map and Du Val double planes, preprint, math.AG n. 0312351."},{"issue":"5-6","key":"8","doi-asserted-by":"publisher","first-page":"799","DOI":"10.4310\/MRL.2004.v11.n6.a7","article-title":"Even sets of four nodes on rational surfaces","volume":"11","author":"Calabri, Alberto","year":"2004","journal-title":"Math. Res. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-2780","issn-type":"print"},{"issue":"2","key":"9","first-page":"99","article-title":"Explicit resolutions of double point singularities of surfaces","volume":"53","author":"Calabri, Alberto","year":"2002","journal-title":"Collect. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-0757","issn-type":"print"},{"key":"10","unstructured":"[Cam] L. Campedelli, Sopra alcuni piani doppi notevoli con curva di diramazione del decimo ordine, Atti Accad. Naz. 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