{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:18:44Z","timestamp":1760242724001,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,4,20]],"date-time":"2016-04-20T00:00:00Z","timestamp":1461110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2013R1A1A2012783"],"award-info":[{"award-number":["2013R1A1A2012783"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we introduce special divisors (root, line, ruling, exceptional system and rational quartic) in smooth rational surfaces and study their correspondences to subpolytopes in Gosset polytopes     k 21    . We also show that the sets of rulings and exceptional systems correspond equivariantly to the vertices of      2  k 1       and      1  k 2       via E-type Weyl action.<\/jats:p>","DOI":"10.3390\/sym8040027","type":"journal-article","created":{"date-parts":[[2016,4,20]],"date-time":"2016-04-20T11:00:40Z","timestamp":1461150040000},"page":"27","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["E-Polytopes in Picard Groups of Smooth Rational Surfaces"],"prefix":"10.3390","volume":"8","author":[{"given":"Jae-Hyouk","family":"Lee","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ewha Womans University, Seoul 03760, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"YongJoo","family":"Shin","sequence":"additional","affiliation":[{"name":"Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 103077, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,4,20]]},"reference":[{"key":"ref_1","unstructured":"Beauville, A. (1978). Surfaces alg\u00e9briques complexes, Ast\u00e9risque, Soci\u00e9t\u00e9 Math\u00e9matique de France."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Demazure, M. (1980). S\u00e9minaire sur les Singularit\u00e9s des Surfaces, Springer-Verlag.","DOI":"10.1007\/BFb0085872"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Dolgachev, I.V. (2012). Classical Algebraic Geometry. A Modern View, Cambridge University Press.","DOI":"10.1017\/CBO9781139084437"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"457","DOI":"10.2307\/2371466","article-title":"The polytope 221, whose twenty-seven vertices correspond to the lines on the general cubic surface","volume":"62","author":"Coxeter","year":"1940","journal-title":"Am. J. Math."},{"key":"ref_5","unstructured":"Manin, Y. (1986). Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Mathematical Library 4. [2nd ed.]. English Translation."},{"key":"ref_6","unstructured":"Friedman, R., and Morgan, J. 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Contractions of del Pezzo Surfaces to \n          \n            \n              \n                \n                  P\n                  2\n                \n              \n            \n          \n         or \n          \n            \n              \n                \n                  P\n                  1\n                \n\t\t\t\t\u00d7\n                \n                  P\n                  1\n                \n              \n            \n          \n        . Rocky Mt. J. Math., in press.","DOI":"10.1216\/RMJ-2016-46-4-1263"},{"key":"ref_13","unstructured":"Lee, J.H., and Shin, Y. (2016). Special Divisor Classes on Blown-up Hirzebruch Surfaces I,II, preprints."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"101","DOI":"10.2140\/pjm.2005.218.101","article-title":"Irreducibility of \u22121-classes on anticanonical rational surfaces and finite generation of the effective monoid","volume":"218","author":"Lahyane","year":"2005","journal-title":"Pac. J. Math."},{"key":"ref_15","unstructured":"Coxeter, H.S.M. (1991). Regular Complex Polytopes, Cambridge University Press. [2nd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/4\/27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:22:36Z","timestamp":1760210556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/4\/27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,20]]},"references-count":15,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,4]]}},"alternative-id":["sym8040027"],"URL":"https:\/\/doi.org\/10.3390\/sym8040027","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2016,4,20]]}}}