{"id":5093,"date":"2010-11-16T16:19:18","date_gmt":"2010-11-16T16:19:18","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5093"},"modified":"2012-02-24T16:30:48","modified_gmt":"2012-02-24T16:30:48","slug":"del-pezzo-surfaces","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5093","title":{"rendered":"del Pezzo surfaces"},"content":{"rendered":"<p>Here are the G-functions (<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/789\/789290098faf37c605cb24e8b208ddcf-ffffff-000000-0.png' alt='I = G_{+s} = e^s \\cdot G' title='I = G_{+s} = e^s \\cdot G' class='latex' \/>)<br \/>\nand regularized quantum periods (<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/844\/844a8a1c77dd5a72aaeb1c9fc942e6d4-ffffff-000000-0.png' alt='I_{reg} = \\hat{G}' title='I_{reg} = \\hat{G}' class='latex' \/>)<br \/>\nfor del Pezzo surfaces.<\/p>\n<h3>The del Pezzo surface of degree 9<\/h3>\n<p>This is the toric variety <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/fc4\/fc40778b711617ef146a3ec76339a0d5-ffffff-000000-0.png' alt='\\PP^2' title='\\PP^2' class='latex' \/>.\u00a0 We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/275\/275fa15fc9dfa3b6a980a81f1e95d613-ffffff-000000-0.png' alt='G(t) = \\sum_k {t^{3k} \\over k!k!k!} = 1+t^3+\\frac{t^6}{8}+\\frac{t^9}{216}+\\cdots' title='G(t) = \\sum_k {t^{3k} \\over k!k!k!} = 1+t^3+\\frac{t^6}{8}+\\frac{t^9}{216}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/208\/2080861ff9268b602473456e780da432-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+6 t^3+90 t^6+1680 t^9+\\cdots' title='\\hat{G}(t) = 1+6 t^3+90 t^6+1680 t^9+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 8, case a<\/h3>\n<p>This is the toric variety <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/ce3\/ce3a4dc55f03066aea89d98807261acd-ffffff-000000-0.png' alt='\\PP^1 \\times \\PP^1' title='\\PP^1 \\times \\PP^1' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/886\/88663638344f6fa163b2ee7baac6cd67-ffffff-000000-0.png' alt='G(t) = \\sum_{k,l} {t^{2k+2l} \\over k!k!l!l!} = 1+2 t^2+\\frac{3 t^4}{2}+\\frac{5 t^6}{9}+\\cdots' title='G(t) = \\sum_{k,l} {t^{2k+2l} \\over k!k!l!l!} = 1+2 t^2+\\frac{3 t^4}{2}+\\frac{5 t^6}{9}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/d91\/d91288139349a51313a306bc37310722-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+4 t^2+36 t^4+400 t^6+\\cdots' title='\\hat{G}(t) = 1+4 t^2+36 t^4+400 t^6+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 8, case b<\/h3>\n<p>This is the toric variety <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/8d7\/8d70c57340e9aa1ea22100d150e22714-ffffff-000000-0.png' alt='\\mathbb{F}_1' title='\\mathbb{F}_1' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/a36\/a36fe99343bfe4cdcd0f6ef5ceb40b3e-ffffff-000000-0.png' alt='G(t) = \\sum_{k,l} {t^{2k+l} \\over k!l!l!(k-l)!} = 1+t^2+t^3+\\frac{t^4}{4}+\\frac{t^5}{2}+\\frac{11 t^6}{72}+\\frac{t^7}{12}+\\cdots' title='G(t) = \\sum_{k,l} {t^{2k+l} \\over k!l!l!(k-l)!} = 1+t^2+t^3+\\frac{t^4}{4}+\\frac{t^5}{2}+\\frac{11 t^6}{72}+\\frac{t^7}{12}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/b8c\/b8c02d4a46e1fed6ec0b6b40b1de2684-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+2 t^2+6 t^3+6 t^4+60 t^5+110 t^6+420 t^7+\\cdots' title='\\hat{G}(t) = 1+2 t^2+6 t^3+6 t^4+60 t^5+110 t^6+420 t^7+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 7<\/h3>\n<p>This is the blow-up of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/fc4\/fc40778b711617ef146a3ec76339a0d5-ffffff-000000-0.png' alt='\\PP^2' title='\\PP^2' class='latex' \/> in two points.  It is a toric variety.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/f30\/f30639c56f6dc1efdee2499d51e8ff62-ffffff-000000-0.png' alt='G(t) = \\sum_{k,l,m} {t^{2k+3l+2m} \\over (l+m)!(k+l)!m!l!k!} =1+2 t^2+t^3+\\frac{3 t^4}{2}+t^5+\\frac{49 t^6}{72}+\\frac{5 t^7}{12}+\\cdots' title='G(t) = \\sum_{k,l,m} {t^{2k+3l+2m} \\over (l+m)!(k+l)!m!l!k!} =1+2 t^2+t^3+\\frac{3 t^4}{2}+t^5+\\frac{49 t^6}{72}+\\frac{5 t^7}{12}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/163\/163d47beaa53cfb175655d82b142d881-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+4 t^2+6 t^3+36 t^4+120 t^5+490 t^6+2100 t^7+\\cdots' title='\\hat{G}(t) = 1+4 t^2+6 t^3+36 t^4+120 t^5+490 t^6+2100 t^7+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 6<\/h3>\n<p>This is the blow-up of <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/fc4\/fc40778b711617ef146a3ec76339a0d5-ffffff-000000-0.png' alt='\\PP^2' title='\\PP^2' class='latex' \/> in three points.  It is a toric variety.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/be8\/be8547a07e20aea1db02e9a3534fc98d-ffffff-000000-0.png' alt='G(t) = \\sum_{k,l,m,n} {t^{k+2l+2m+n} \\over (-k+m+n)!(k+l-n)!n!m!l!k!} =1+3 t^2+2 t^3+\\frac{15 t^4}{4}+3 t^5+\\frac{17 t^6}{6}+2 t^7+\\cdots' title='G(t) = \\sum_{k,l,m,n} {t^{k+2l+2m+n} \\over (-k+m+n)!(k+l-n)!n!m!l!k!} =1+3 t^2+2 t^3+\\frac{15 t^4}{4}+3 t^5+\\frac{17 t^6}{6}+2 t^7+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/631\/6314cb52ed0f1daec974590713632dc6-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+6 t^2+12 t^3+90 t^4+360 t^5+2040 t^6+10080 t^7+\\cdots' title='\\hat{G}(t) = 1+6 t^2+12 t^3+90 t^4+360 t^5+2040 t^6+10080 t^7+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 5<\/h3>\n<p>This is a hypersurface of bidegree <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/369\/369a154d5347e70114104fec91cf72fe-ffffff-000000-0.png' alt='(1,2)' title='(1,2)' class='latex' \/> in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/d41\/d41f286daa53cd89103600b4e796d684-ffffff-000000-0.png' alt='\\PP^1 \\times \\PP^2' title='\\PP^1 \\times \\PP^2' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/2e9\/2e9c42b48ab6f3423cc4784a20fe4277-ffffff-000000-0.png' alt='G_{+3}(t) = \\sum_{k,l} t^{k+l} {(k+2l)! \\over k!k!l!l!l!} = 1+3 t+\\frac{19 t^2}{2}+\\frac{49 t^3}{2}+\\frac{417 t^4}{8}+\\frac{3751 t^5}{40}+\\frac{104959 t^6}{720}+\\frac{334769 t^7}{1680}+\\cdots' title='G_{+3}(t) = \\sum_{k,l} t^{k+l} {(k+2l)! \\over k!k!l!l!l!} = 1+3 t+\\frac{19 t^2}{2}+\\frac{49 t^3}{2}+\\frac{417 t^4}{8}+\\frac{3751 t^5}{40}+\\frac{104959 t^6}{720}+\\frac{334769 t^7}{1680}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/dfd\/dfd598fd227af11529e1a4e10a83c8cb-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+10 t^2+30 t^3+270 t^4+1560 t^5+11350 t^6+77700 t^7+\\cdots' title='\\hat{G}(t) = 1+10 t^2+30 t^3+270 t^4+1560 t^5+11350 t^6+77700 t^7+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 4<\/h3>\n<p>This is a (2,2) complete intersection in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/e68\/e68461e5992f8ae959bc527dfa5f8294-ffffff-000000-0.png' alt='\\PP^4' title='\\PP^4' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/2e3\/2e3751929db2315afeae304e0756b710-ffffff-000000-0.png' alt='G_{+4}(t) = \\sum_k t^{k} {(2k)!(2k)! \\over k!k!k!k!k!} = 1+4 t+18 t^2+\\frac{200 t^3}{3}+\\frac{1225 t^4}{6}+\\frac{2646 t^5}{5}+\\frac{5929 t^6}{5}+\\frac{81796 t^7}{35}+\\cdots' title='G_{+4}(t) = \\sum_k t^{k} {(2k)!(2k)! \\over k!k!k!k!k!} = 1+4 t+18 t^2+\\frac{200 t^3}{3}+\\frac{1225 t^4}{6}+\\frac{2646 t^5}{5}+\\frac{5929 t^6}{5}+\\frac{81796 t^7}{35}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/a30\/a3046de57b2086979b74ceab777db996-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+20 t^2+96 t^3+1188 t^4+10560 t^5+111440 t^6+1142400 t^7+\\cdots' title='\\hat{G}(t) = 1+20 t^2+96 t^3+1188 t^4+10560 t^5+111440 t^6+1142400 t^7+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 3<\/h3>\n<p>This is the cubic surface in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/805\/80578734ac32d4a21437d3b40ee514e4-ffffff-000000-0.png' alt='\\PP^3' title='\\PP^3' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/45e\/45e21e537fe4b7b24a13d503778774dc-ffffff-000000-0.png' alt='G_{+6}(t) = \\sum_k t^{k} {(3k)!\\over k!k!k!k!} = 1+6 t+45 t^2+280 t^3+\\frac{5775 t^4}{4}+\\frac{63063 t^5}{10}+\\frac{119119 t^6}{5}+\\frac{554268 t^7}{7}+\\cdots' title='G_{+6}(t) = \\sum_k t^{k} {(3k)!\\over k!k!k!k!} = 1+6 t+45 t^2+280 t^3+\\frac{5775 t^4}{4}+\\frac{63063 t^5}{10}+\\frac{119119 t^6}{5}+\\frac{554268 t^7}{7}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/ed2\/ed2512a4eb61900fc4c7052196e47a36-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+54 t^2+492 t^3+9882 t^4+158760 t^5+2879640 t^6+51982560 t^7+\\cdots' title='\\hat{G}(t) = 1+54 t^2+492 t^3+9882 t^4+158760 t^5+2879640 t^6+51982560 t^7+\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 2<\/h3>\n<p>This is the quartic surface in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/4eb\/4eba3471c0fe62909f2a85052f9ef088-ffffff-000000-0.png' alt='\\PP(1,1,1,2)' title='\\PP(1,1,1,2)' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/927\/927319f2296f6970d1f368002c94b249-ffffff-000000-0.png' alt='G_{+12}(t) = \\sum_k t^{k} {(4k)!\\over k!k!k!(2k)!} = 1+12 t+210 t^2+3080 t^3+\\frac{75075 t^4}{2}+\\frac{1939938 t^5}{5}+\\frac{52055003 t^6}{15}+\\frac{191222460 t^7}{7}+\\cdots' title='G_{+12}(t) = \\sum_k t^{k} {(4k)!\\over k!k!k!(2k)!} = 1+12 t+210 t^2+3080 t^3+\\frac{75075 t^4}{2}+\\frac{1939938 t^5}{5}+\\frac{52055003 t^6}{15}+\\frac{191222460 t^7}{7}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/2e4\/2e48b72dfe071833b14eb6bdcf05f2a7-ffffff-000000-0.png' alt='\\hat{G}(t) = 1+276 t^2+6816 t^3+314532 t^4+12853440 t^5+569409360 t^6+25533244800 t^7 +\\cdots' title='\\hat{G}(t) = 1+276 t^2+6816 t^3+314532 t^4+12853440 t^5+569409360 t^6+25533244800 t^7 +\\cdots' class='latex' \/><\/p>\n<h3>The del Pezzo surface of degree 1<\/h3>\n<p>This is the sextic surface in <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/957\/95792d9316d80ba178cd384b6442b0c9-ffffff-000000-0.png' alt='\\PP(1,1,2,3)' title='\\PP(1,1,2,3)' class='latex' \/>.  We have:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/54d\/54d2c7bf356c0867e1605e46ac8e27b2-ffffff-000000-0.png' alt='G_{+60}(t) = \\sum_k t^{k} {(6k)!\\over k!k!(2k)!(3k)!} = 1+60 t+6930 t^2+680680 t^3+\\frac{111546435 t^4}{2}+3881815938 t^5+233987238485 t^6+\\frac{86928646722060 t^7}{7}+\\cdots' title='G_{+60}(t) = \\sum_k t^{k} {(6k)!\\over k!k!(2k)!(3k)!} = 1+60 t+6930 t^2+680680 t^3+\\frac{111546435 t^4}{2}+3881815938 t^5+233987238485 t^6+\\frac{86928646722060 t^7}{7}+\\cdots' class='latex' \/><br \/>\nand the regularized quantum period is:<br \/>\n<img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/d32\/d32dac74535e2b0e392a3bc9cd4982ea-ffffff-000000-0.png' alt='\\hat{G}(t) = 1 + 10260 t^2 + 2021280 t^3 + 618874020 t^4 + 184450426560 t^5 + 57876331467600 t^6 + 18570232920355200 t^7+\\cdots' title='\\hat{G}(t) = 1 + 10260 t^2 + 2021280 t^3 + 618874020 t^4 + 184450426560 t^5 + 57876331467600 t^6 + 18570232920355200 t^7+\\cdots' class='latex' \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here are the G-functions () and regularized quantum periods () for del Pezzo surfaces. The del Pezzo surface of degree 9 This is the toric variety .\u00a0 We have: and the regularized quantum period is: The del Pezzo surface of degree 8, case a This is the toric variety . We have: and the regularized [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-5093","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5093","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5093"}],"version-history":[{"count":23,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5093\/revisions"}],"predecessor-version":[{"id":5626,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5093\/revisions\/5626"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5093"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5093"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5093"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}