{"id":277,"date":"2010-09-09T15:04:34","date_gmt":"2010-09-09T15:04:34","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=277"},"modified":"2012-02-27T12:56:31","modified_gmt":"2012-02-27T12:56:31","slug":"fano-3-folds","status":"publish","type":"page","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=277","title":{"rendered":"Fano Varieties"},"content":{"rendered":"

Legend for period sequences:
\nn\/a – known, and not one of 165 from Minkowski list, [7 entries]
\n# – known to be number # from Minkowski list, [91 entries]
\nA# – known to be number # from Minkowski list, computed with A\/NA corr., [5 entries]
\n?*# – unknown, but STD ansatz predicts it is number #, [1 entry]
\n? – unknown and no guess. [0 entries]<\/p>\n

Rank 1 Fano 3-folds<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Name<\/th>\nFano index<\/th>\nPeriod sequence<\/th>\nConstruction<\/th>\n<\/tr>\n
<\/a>\\PP^3<\/td>\n4<\/td>\n12<\/td>\nprojective 3-space<\/td>\n<\/tr>\n
<\/a>Q^3<\/td>\n3<\/td>\n4<\/td>\nthe quadric 3-fold<\/td>\n<\/tr>\n
<\/a>B_5<\/td>\n2<\/td>\n14<\/td>\na linear section of \\mathrm{Gr}(2,5)<\/td>\n<\/tr>\n
<\/a>B_4<\/td>\n2<\/td>\n2<\/td>\na (2,2) complete intersection in \\PP^5<\/td>\n<\/tr>\n
<\/a>B_3<\/td>\n2<\/td>\n0<\/td>\nthe cubic 3-fold<\/td>\n<\/tr>\n
<\/a>B_2<\/td>\n2<\/td>\n1<\/td>\na quartic in \\PP(1,1,1,1,2)<\/td>\n<\/tr>\n
<\/a>B_1<\/td>\n2<\/td>\nn\/a<\/td>\na sextic in \\PP(1,1,1,2,3)<\/td>\n<\/tr>\n
<\/a>V_2<\/td>\n1<\/td>\nn\/a<\/td>\na sextic in \\PP(1,1,1,1,3)<\/td>\n<\/tr>\n
<\/a>V_4<\/td>\n1<\/td>\n15<\/td>\na quartic in \\PP^4<\/td>\n<\/tr>\n
<\/a>V_6<\/td>\n1<\/td>\n19<\/td>\na (2,3) complete intersection in \\PP^5<\/td>\n<\/tr>\n
<\/a>V_8<\/td>\n1<\/td>\n5<\/td>\na (2,2,2) complete intersection in \\PP^6<\/td>\n<\/tr>\n
<\/a>V_{10}<\/td>\n1<\/td>\n9<\/td>\nsection of Gr(2,5) by intersection of quadric and linear space<\/td>\n<\/tr>\n
<\/a>V_{12}<\/td>\n1<\/td>\n7<\/td>\nlinear section of spinor variety OGr(5,10)<\/td>\n<\/tr>\n
<\/a>V_{14}<\/td>\n1<\/td>\n18<\/td>\nlinear section of Gr(2,6)<\/td>\n<\/tr>\n
<\/a>V_{16}<\/td>\n1<\/td>\n20<\/td>\nlinear section of LG(3,6)<\/td>\n<\/tr>\n
<\/a>V_{18}<\/td>\n1<\/td>\n10<\/td>\nlinear section of G_2\/P<\/td>\n<\/tr>\n
<\/a>V_{22}<\/td>\n1<\/td>\n17<\/td>\nsection of triple universal bundle on Gr(3,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Rank 2 Fano 3-folds<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Number<\/th>\n(-K_X)^3<\/th>\nPeriod sequence<\/th>\nDescription<\/th>\n<\/tr>\n
<\/a>1<\/td>\n4<\/td>\nn\/a<\/td>\nthe blow-up of V_1 with centre an elliptic curve which is the intersection of two members of |{-{1\/2}} K_{V_1}|<\/td>\n<\/tr>\n
<\/a>2<\/td>\n6<\/td>\nn\/a<\/td>\na double cover of \\PP^1 \\times \\PP^2 branched along a divisor of bidegree (2,4)<\/td>\n<\/tr>\n
<\/a>3<\/td>\n8<\/td>\nn\/a<\/td>\nthe blow up of V_2 with centre an elliptic curve which is the intersection of two elements of |-1\/2K_{V_2}|<\/td>\n<\/tr>\n
<\/a>4<\/td>\n10<\/td>\n49<\/td>\nthe blow up of \\PP^3 with centre an intersection of two cubics<\/td>\n<\/tr>\n
<\/a>5<\/td>\n12<\/td>\n34<\/td>\nthe blow up of V_3 with centre a plane cubic on it<\/td>\n<\/tr>\n
<\/a>6<\/td>\n12<\/td>\n11<\/td>\na divisor of bidegree (2,2) in \\PP^2 \\times \\PP^2<\/td>\n<\/tr>\n
<\/a>7<\/td>\n14<\/td>\n51<\/td>\nthe blow up of a quadric Q with centre the intersection of two members of |\\cO_Q(2)|<\/td>\n<\/tr>\n
<\/a>8<\/td>\n14<\/td>\n26<\/td>\na double cover of V_7 with branch locus a member B of |-K_{V_7}| such that B\\cap D is smooth, where D is the exceptional divisor of the blow-up V_7 \\to \\PP^3<\/td>\n<\/tr>\n
<\/a>9<\/td>\n16<\/td>\n62<\/td>\nthe blow up of \\PP^3 in a curve \\Gamma of degree 7 and genus 5<\/td>\n<\/tr>\n
<\/a>10<\/td>\n16<\/td>\n40<\/td>\nthe blow up of V_4 with centre an elliptic curve which is the intersection of two hyperplane sections<\/td>\n<\/tr>\n
<\/a>11<\/td>\n18<\/td>\n56<\/td>\nthe blow up of V_3 with centre a line on it<\/td>\n<\/tr>\n
<\/a>12<\/td>\n20<\/td>\n13<\/td>\nthe blow up of \\PP^3 in a curve \\Gamma of degree 6 and genus 3<\/td>\n<\/tr>\n
<\/a>13<\/td>\n20<\/td>\n52<\/td>\nthe blow-up of a 3-dimensional quadric Q in a curve \\Gamma of genus 2 and degree 6<\/td>\n<\/tr>\n
<\/a>14<\/td>\n20<\/td>\nA 39<\/td>\nthe blow-up of V_5 with center an elliptic curve which is an intersection of two hyperplane sections<\/td>\n<\/tr>\n
<\/a>15<\/td>\n22<\/td>\n35<\/td>\nthe blow-up of \\PP^3 with center the intersection of a quadric and a cubic<\/td>\n<\/tr>\n
<\/a>16<\/td>\n22<\/td>\n59<\/td>\nthe blow-up of V_4 \\subset \\PP^5 with center a conic on it<\/td>\n<\/tr>\n
<\/a>17<\/td>\n24<\/td>\nA 38<\/td>\nthe blow-up of a 3-dimensional quadric Q with center an elliptic curve of degree 5 on it<\/td>\n<\/tr>\n
<\/a>18<\/td>\n24<\/td>\n60<\/td>\na double cover of \\PP^1 \\times \\PP^2 with branch locus a divisor of bidegree (2,2)<\/td>\n<\/tr>\n
<\/a>19<\/td>\n26<\/td>\n55<\/td>\nthe blow-up of V_4 with center a line on it<\/td>\n<\/tr>\n
<\/a>20<\/td>\n26<\/td>\nA 46<\/td>\nthe blow-up of V_5 \\subset \\PP^6 with center a twisted cubic on it<\/td>\n<\/tr>\n
<\/a>21<\/td>\n28<\/td>\nA 8<\/td>\nthe blow-up of Q \\subset \\PP^4 with center a twisted quartic on it (a smooth rational curve of degree 4 which spans \\PP^4)<\/td>\n<\/tr>\n
<\/a>22<\/td>\n30<\/td>\nA 50<\/td>\nthe blow-up of V_5 with center a conic on it<\/td>\n<\/tr>\n
<\/a>23<\/td>\n30<\/td>\n29<\/td>\nthe blow-up of a quadric with center an intersection of A \\in |\\cO(1)| and B \\in |\\cO(2)|<\/td>\n<\/tr>\n
<\/a>24<\/td>\n30<\/td>\n66<\/td>\nA divisor of bidegree (1,2) on \\PP^2 \\times \\PP^2<\/td>\n<\/tr>\n
<\/a>25<\/td>\n32<\/td>\n28<\/td>\nThe blow up of \\PP^3 with centre an elliptic curve which is the complete intersection of two quadrics<\/td>\n<\/tr>\n
<\/a>26<\/td>\n34<\/td>\n?*47<\/td>\nthe blow up of V_5 with center a line on it<\/td>\n<\/tr>\n
<\/a>27<\/td>\n38<\/td>\n61<\/td>\nthe blow up of \\PP^3 with center a twisted cubic<\/td>\n<\/tr>\n
<\/a>28<\/td>\n40<\/td>\n33<\/td>\nthe blow-up of \\PP^3 with centre a plane cubic<\/td>\n<\/tr>\n
<\/a>29<\/td>\n40<\/td>\n42<\/td>\nthe blow-up of a quadric 3-fold Q with centre a conic on it<\/td>\n<\/tr>\n
<\/a>30<\/td>\n46<\/td>\n70<\/td>\nthe blow-up of \\PP^3 with center a conic<\/td>\n<\/tr>\n
<\/a>31<\/td>\n46<\/td>\n48<\/td>\nthe blow-up of a quadric 3-fold Q with center a line on it<\/td>\n<\/tr>\n
<\/a>32<\/td>\n48<\/td>\n6<\/td>\na divisor W on \\PP^2 \\times \\PP^2 of bidegree (1,1)<\/td>\n<\/tr>\n
<\/a>33<\/td>\n54<\/td>\n54<\/td>\nthe blow-up of \\PP^3 with center a line<\/td>\n<\/tr>\n
<\/a>34<\/td>\n54<\/td>\n44<\/td>\nX = \\PP^1 \\times \\PP^2<\/td>\n<\/tr>\n
<\/a>35<\/td>\n56<\/td>\n30<\/td>\nV_7, which is the blow-up of \\PP^3 at a point<\/td>\n<\/tr>\n
<\/a>36<\/td>\n62<\/td>\n58<\/td>\nthe scroll \\PP(\\cO \\oplus \\cO(2)) over \\PP^2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Rank 3 Fano 3-folds<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Number<\/th>\n(-K_X)^3<\/th>\nPeriod sequence<\/th>\nDescription<\/th>\n<\/tr>\n
<\/a>1<\/td>\n12<\/td>\n22<\/td>\na double cover of \\PP^1 \\times \\PP^1 \\times \\PP^1 with branch locus a divisor of tridegree (2,2,2)<\/td>\n<\/tr>\n
<\/a>2<\/td>\n14<\/td>\n97<\/td>\na member of |L^{\\otimes 2} \\otimes \\cO(2,3)| on the \\PP^2-bundle \\PP(\\cO\\oplus\\cO(-1,-1)^{\\oplus 2}) over \\PP^1 \\times \\PP^1, where L is the tautological line bundle<\/td>\n<\/tr>\n
<\/a>3<\/td>\n18<\/td>\n31<\/td>\na divisor on \\PP^1 \\times \\PP^1 \\times \\PP^2 of tridegree (1,1,2)<\/td>\n<\/tr>\n
<\/a>4<\/td>\n18<\/td>\n151<\/td>\nthe blow-up of Y (rank 2 table, number 18) with center a smooth fiber of Y \\to \\PP^1 \\times \\PP^2 \\to \\PP^2, where the first map is the double cover and the second map is the projection<\/td>\n<\/tr>\n
<\/a>5<\/td>\n20<\/td>\n109<\/td>\nthe blow-up of \\PP^1 \\times \\PP^2 with center a curve of bidegree (5,2) that projects isomorphically to a conic in \\PP^2<\/td>\n<\/tr>\n
<\/a>6<\/td>\n22<\/td>\n146<\/td>\nthe blow-up of \\PP^3 with center a disjoint union of a line and an elliptic curve of degree 4<\/td>\n<\/tr>\n
<\/a>7<\/td>\n24<\/td>\n36<\/td>\nthe blow-up X of W with center an elliptic curve which is a complete intersection of two members of |-{1 \\over 2} K_W|<\/td>\n<\/tr>\n
<\/a>8<\/td>\n24<\/td>\n85<\/td>\na member of the linear system |p_1^\\star g^\\star \\cO(1) \\otimes p_2^\\star \\cO(2)| on \\mathbb{F}_1 \\times \\PP^2, where p_1, p_2 are the projections and g:\\mathbb{F}_1 \\to \\PP^2 is the blow-up<\/td>\n<\/tr>\n
<\/a>9<\/td>\n26<\/td>\n68<\/td>\nthe blow-up of the cone W_4 \\subset \\PP^6 over the Veronese surface R_4 \\subset \\PP^5 with center a disjoint union of the vertex and a quartic in R_4 \\cong \\PP^2<\/td>\n<\/tr>\n
<\/a>10<\/td>\n26<\/td>\n67<\/td>\nthe blow-up of Q \\subset \\PP^4 with center a disjoint union of two conics on it<\/td>\n<\/tr>\n
<\/a>11<\/td>\n28<\/td>\n107<\/td>\nthe blow-up X of V_7 with center a complete intersection of two general members of {-1\/2} K_{V_7}<\/td>\n<\/tr>\n
<\/a>12<\/td>\n28<\/td>\n144<\/td>\nthe blow-up of \\PP^3 with centre a disjoint union of a twisted cubic and a line<\/td>\n<\/tr>\n
<\/a>13<\/td>\n30<\/td>\n16<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>14<\/td>\n32<\/td>\n148<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>15<\/td>\n32<\/td>\n112<\/td>\nthe blow-up X of a quadric Q \\subset \\PP^4 with center the disjoint union of a line on Q and a conic on Q<\/td>\n<\/tr>\n
<\/a>16<\/td>\n34<\/td>\n119<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>17<\/td>\n36<\/td>\n37<\/td>\na smooth divisor on \\PP^1 \\times \\PP^1 \\times \\PP^2 of tridegree (1,1,1)<\/td>\n<\/tr>\n
<\/a>18<\/td>\n36<\/td>\n160<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>19<\/td>\n38<\/td>\n57<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>20<\/td>\n38<\/td>\n63<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>21<\/td>\n38<\/td>\n98<\/td>\nthe blow-up of \\PP^1 \\times \\PP^2 along a curve of bidegree (2,1)<\/td>\n<\/tr>\n
<\/a>22<\/td>\n40<\/td>\n152<\/td>\nthe blow-up of \\PP^1 \\times \\PP^2 along a curve of bidegree (0,2), that is a conic in \\{t\\}\\times \\PP^2<\/td>\n<\/tr>\n
<\/a>23<\/td>\n42<\/td>\n158<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>24<\/td>\n42<\/td>\n86<\/td>\nthe fiber product X = W \\times_{\\PP^2} \\mathbb{F}_1 where W is a (1,1) hypersurface in \\PP^2 \\times \\PP^2<\/td>\n<\/tr>\n
<\/a>25<\/td>\n44<\/td>\n41<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>26<\/td>\n46<\/td>\n113<\/td>\nthe blow-up X of \\PP^3 with center a disjoint union of a point and a line<\/td>\n<\/tr>\n
<\/a>27<\/td>\n48<\/td>\n21<\/td>\nX = \\PP^1 \\times \\PP^1 \\times \\PP^1<\/td>\n<\/tr>\n
<\/a>28<\/td>\n48<\/td>\n90<\/td>\nX = \\PP^1 \\times \\mathbb{F}_1<\/td>\n<\/tr>\n
<\/a>29<\/td>\n50<\/td>\n163<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>30<\/td>\n50<\/td>\n84<\/td>\nthe blow-up<\/td>\n<\/tr>\n
<\/a>31<\/td>\n52<\/td>\n53<\/td>\nthe blow-up<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Rank 4 Fano 3-folds<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Number<\/th>\n(-K_X)^3<\/th>\nPeriod sequence<\/th>\nDescription<\/th>\n<\/tr>\n
<\/a>1<\/td>\n24<\/td>\n3<\/td>\na smooth divisor on \\PP^1 \\times \\PP^1 \\times \\PP^1 \\times \\PP^1 of polydegree (1,1,1,1)<\/td>\n<\/tr>\n
<\/a>2<\/td>\n28<\/td>\n32<\/td>\n<\/td>\n<\/tr>\n
<\/a>3<\/td>\n30<\/td>\n122<\/td>\n<\/td>\n<\/tr>\n
<\/a>4<\/td>\n32<\/td>\n103<\/td>\n<\/td>\n<\/tr>\n
<\/a>5<\/td>\n32<\/td>\n147<\/td>\n<\/td>\n<\/tr>\n
<\/a>6<\/td>\n34<\/td>\n65<\/td>\n<\/td>\n<\/tr>\n
<\/a>7<\/td>\n36<\/td>\n69<\/td>\n<\/td>\n<\/tr>\n
<\/a>8<\/td>\n38<\/td>\n105<\/td>\n<\/td>\n<\/tr>\n
<\/a>9<\/td>\n40<\/td>\n102<\/td>\n<\/td>\n<\/tr>\n
<\/a>10<\/td>\n42<\/td>\n142<\/td>\n<\/td>\n<\/tr>\n
<\/a>11<\/td>\n44<\/td>\n93<\/td>\n<\/td>\n<\/tr>\n
<\/a>12<\/td>\n46<\/td>\n150<\/td>\n<\/td>\n<\/tr>\n
<\/a>13<\/td>\n26<\/td>\n88<\/td>\nblowup of curve of tridegree (1,1,3) on \\PP^1 \\times \\PP^1 \\times \\PP^1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Rank 5 Fano 3-folds<\/h1>\n\n\n\n\n\n\n
Number<\/th>\n(-K_X)^3<\/th>\nPeriod sequence<\/th>\nDescription<\/th>\n<\/tr>\n
<\/a>1<\/td>\n28<\/td>\n114<\/td>\n<\/td>\n<\/tr>\n
<\/a>2<\/td>\n36<\/td>\n87<\/td>\n“smooth toric, but not 5-3”<\/td>\n<\/tr>\n
<\/a>3<\/td>\n36<\/td>\n43<\/td>\n\\PP^1 \\times S_6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Fano 3-folds with B_2 > 5<\/h1>\n\n\n\n\n\n\n\n\n
Picard number<\/th>\n(-K_X)^3<\/th>\nPeriod sequence<\/th>\nDescription<\/th>\n<\/tr>\n
<\/a>6<\/td>\n30<\/td>\n64<\/td>\n\\PP^1 \\times S_5<\/td>\n<\/tr>\n
<\/a>7<\/td>\n24<\/td>\n71<\/td>\n\\PP^1 \\times S_4<\/td>\n<\/tr>\n
<\/a>8<\/td>\n18<\/td>\n45<\/td>\n\\PP^1 \\times S_3<\/td>\n<\/tr>\n
<\/a>9<\/td>\n12<\/td>\nn\/a<\/td>\n\\PP^1 \\times S_2<\/td>\n<\/tr>\n
<\/a>10<\/td>\n6<\/td>\nn\/a<\/td>\n\\PP^1 \\times S_1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"

Legend for period sequences: n\/a – known, and not one of 165 from Minkowski list, [7 entries] # – known to be number # from Minkowski list, [91 entries] A# – known to be number # from Minkowski list, computed with A\/NA corr., [5 entries] ?*# – unknown, but STD ansatz predicts it is number […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-277","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/277","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/types\/page"}],"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=277"}],"version-history":[{"count":47,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/277\/revisions"}],"predecessor-version":[{"id":5661,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/277\/revisions\/5661"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=277"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}