{"id":5118,"date":"2010-11-22T14:10:39","date_gmt":"2010-11-22T14:10:39","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5118"},"modified":"2010-12-03T12:17:23","modified_gmt":"2010-12-03T12:17:23","slug":"pictures","status":"publish","type":"post","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?p=5118","title":{"rendered":"Pictures"},"content":{"rendered":"<p>Here are some pictures of the anticanonical surfaces inside some of our Fano 3-folds.\u00a0 In each case we express the Fano <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/021\/02129bb861061d1a052c592e2dc6b383-ffffff-000000-0.png' alt='X' title='X' class='latex' \/> as a complete intersection in a toric variety <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/57c\/57cec4137b614c87cb4e24a3d003a3e0-ffffff-000000-0.png' alt='Y' title='Y' class='latex' \/>, choose an affine chart on <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/57c\/57cec4137b614c87cb4e24a3d003a3e0-ffffff-000000-0.png' alt='Y' title='Y' class='latex' \/>, take the intersection of an anticanonical hypersurface with that affine chart, and then take a generic projection of the resulting surface into affine 3-space.\u00a0 Thanks to Andrew MacPherson for the images.<\/p>\n<table>\n<tbody>\n<tr>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5130\" title=\"1-V6-1\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-1-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-1-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-1-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-1.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5128\" title=\"1-V6-2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-2-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-2-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-2-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-V6-2.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2\">the rank-1 Fano 3-fold <img src='http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/latex\/9f0\/9f054e777730e7d9d7007d94d5c97111-ffffff-000000-0.png' alt='V_6' title='V_6' class='latex' \/><\/th>\n<\/tr>\n<tr>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5124\" title=\"1-B3-1\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-1-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-1-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-1-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-1.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5123\" title=\"1-B3-2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-2-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-2-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-2-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/1-B3-2.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2&quot;\">The cubic 3-fold<\/th>\n<\/tr>\n<tr>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5122\" title=\"3-1-1\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-1-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-1-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-1-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-1.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5121\" title=\"3-1-2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-2-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-2-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-2-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-1-2.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2&quot;\">Rank 3 #1<\/th>\n<\/tr>\n<tr>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5129\" title=\"3-4-1\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-1-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-1-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-1-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-1.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5127\" title=\"3-4-2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-2-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-2-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-2-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-2.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2&quot;\">Rank 3, #4<\/th>\n<\/tr>\n<tr>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-1.jpg\"><\/a><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5125\" title=\"4-2-1\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-1-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-1-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-1-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-1.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<td><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/3-4-2.jpg\"><\/a><a href=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-2.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5126\" title=\"4-2-2\" src=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-2-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-2-300x300.jpg 300w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-2-150x150.jpg 150w, http:\/\/coates.ma.ic.ac.uk\/fanosearch\/wp-content\/uploads\/2010\/11\/4-2-2.jpg 600w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2&quot;\">Rank 4, #2<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Here are some pictures of the anticanonical surfaces inside some of our Fano 3-folds.\u00a0 In each case we express the Fano as a complete intersection in a toric variety , choose an affine chart on , take the intersection of an anticanonical hypersurface with that affine chart, and then take a generic projection of the [&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-5118","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\/5118","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=5118"}],"version-history":[{"count":13,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5118\/revisions"}],"predecessor-version":[{"id":5143,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/posts\/5118\/revisions\/5143"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5118"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}