{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:41:39Z","timestamp":1776836499578,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"339","license":[{"start":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T00:00:00Z","timestamp":1694476800000},"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>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a del Pezzo surface of degree one over an algebraically closed field, and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript upper X\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mi>X<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">K_X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    its canonical divisor. The morphism\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c6\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varphi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    induced by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue minus 2 upper K Subscript upper X 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:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msub>\n                              <mml:mi>K<\/mml:mi>\n                              <mml:mi>X<\/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_X|<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    realizes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as a double cover of a cone in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper P cubed\">\n                        <mml:semantics>\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:annotation encoding=\"application\/x-tex\">\\mathbb {P}^3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , ramified over a smooth sextic curve. The surface\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    contains 240 exceptional curves. We prove the following statements. For a point\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P\">\n                        <mml:semantics>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">P<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on the ramification curve of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c6\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varphi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , at most sixteen exceptional curves contain\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P\">\n                        <mml:semantics>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">P<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in characteristic\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                    , and at most ten in all other characteristics. Moreover, for a point\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q\">\n                        <mml:semantics>\n                          <mml:mi>Q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">Q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    outside the ramification curve, at most twelve exceptional curves contain\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q\">\n                        <mml:semantics>\n                          <mml:mi>Q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">Q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in characteristic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and at most ten in all other characteristics. We show that these upper bounds are sharp, except possibly in characteristic 5 outside the ramification curve.\n                  <\/p>","DOI":"10.1090\/mcom\/3779","type":"journal-article","created":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T10:34:32Z","timestamp":1662978872000},"page":"451-481","source":"Crossref","is-referenced-by-count":0,"title":["Concurrent lines on del Pezzo surfaces of degree one"],"prefix":"10.1090","volume":"92","author":[{"given":"Ronald","family":"van Luijk","sequence":"first","affiliation":[]},{"given":"Rosa","family":"Winter","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,9,12]]},"reference":[{"issue":"3-4","key":"1","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. I. The user language","volume":"24","author":"Bosma, Wieb","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"1197","DOI":"10.1080\/00927879908826489","article-title":"Lines on del Pezzo surfaces with \ud835\udc3e\u00b2_{\ud835\udc46}=1 in characteristic \u22602","volume":"27","author":"Cragnolini, P.","year":"1999","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"key":"3","unstructured":"[Coda] Magma code, Proposition 4.6, \\url{http:\/\/www.rosa-winter.com\/MagmaConcurrentLines.txt}."},{"key":"4","unstructured":"[Codb] Magma code, Four polynomials Proposition 4.6, \\url{http:\/\/www.rosa-winter.com\/FourPolynomials.txt}."},{"key":"5","doi-asserted-by":"crossref","unstructured":"[Dem80] M. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"issue":"7","key":"12","doi-asserted-by":"publisher","first-page":"729","DOI":"10.2140\/ant.2009.3.729","article-title":"Cox rings of degree one del Pezzo surfaces","volume":"3","author":"Testa, Damiano","year":"2009","journal-title":"Algebra Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1937-0652","issn-type":"print"},{"key":"13","unstructured":"[Win21] R. Winter, Geometry and arithmetic of del Pezzo surfaces of degree 1, Ph.D. Thesis, Universiteit Leiden, 2021, \\url{https:\/\/scholarlypublications.universiteitleiden.nl\/handle\/1887\/138942}."},{"issue":"6","key":"14","doi-asserted-by":"publisher","first-page":"1965","DOI":"10.1007\/s00373-021-02315-8","article-title":"The action of the Weyl group on the \ud835\udc38\u2088 root system","volume":"37","author":"Winter, Rosa","year":"2021","journal-title":"Graphs Combin.","ISSN":"https:\/\/id.crossref.org\/issn\/0911-0119","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03779-0\/S0025-5718-2022-03779-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:47:49Z","timestamp":1776833269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-339\/S0025-5718-2022-03779-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,12]]},"references-count":14,"journal-issue":{"issue":"339","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["S0025-5718-2022-03779-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3779","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2022,9,12]]}}}