{"id":228,"date":"2010-07-16T10:33:08","date_gmt":"2010-07-16T10:33:08","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=228"},"modified":"2012-02-27T12:57:24","modified_gmt":"2012-02-27T12:57:24","slug":"rank-3-fanos","status":"publish","type":"page","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=228","title":{"rendered":"Rank 3 Fano 3-folds"},"content":{"rendered":"
    \n
  1. a double cover of \\PP^1 \\times \\PP^1 \\times \\PP^1 branched along a divisor of tridegree (2,2,2).\u00a0 This is a hypersurface of type 2L+2M+2N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & t_0 & t_1 & u_0 & u_1 & y & \\\\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & L \\\\ 0& 0 & 1 & 1 & 0 & 0 & 1 & M\\\\ 0 & 0 & 0 & 0& 0 &0 & 1 & N \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(2l+2m+2n)! \\over l! l!m! m!n!n!(l+m+n)!}
    \nand regularizing gives period sequence 22:
    \n1+54 t^2+672 t^3+15642 t^4+336960 t^5 + \\cdots
    \nNote that this is a G-Fano.<\/li>\n
  2. a member of |L^{\\otimes 2} \\otimes_{\\cO_{\\PP^1 \\times \\PP^1}} \\cO(2,3)| on the \\PP^2-bundle \\PP(\\cO \\oplus \\cO(-1,-1)^{\\oplus 2}) over \\PP^1 \\times \\PP^1 such that X \\cap Y is irreducible, where L is the tautological line bundle and Y is a member of |L|.\u00a0 As discussed here<\/a>, this is period sequence 97.\n

    We do not understand this variety<\/strong>: in particular it does not seem to be Fano.Write F for the ambient \\PP^2-bundle, which is the toric variety with weight data:<\/span>
    \n \\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & s_0 & s_1 & t & \\\\ 1 & 1 & 0 & 0 & -1 & -1 & 0 & A \\\\ 0& 0 & 1 & 1 & -1 & -1 & 0 & B\\\\ 0 & 0 & 0 & 0& 1 &1 & 1 & C \\end{array} <\/span>
    \n The line bundle L is {-A}-B+C, so X is a member of |B+2C| and -K_X = C-B.\u00a0 The equation defining X takes the form:<\/span>
    \n t^2 a_1(y) + s_0 t b_{1,2}(x,y) + s_1 t c_{1,2}(x,y) + s_0^2 d_{2,3}(x,y) + s_0 s_1 e_{2,3}(x,y) + s_1^2 f_{1,2}(x,y) = 0<\/span>
    \n where a_1(y) a linear function of y_0, y_1; b_{1,2}(x,y) and c_{1,2}(x,y) are homogeneous functions of x_0, x_1, y_0, y_1 of bidegrees (1,2) in the x_i and y_j; and c_{2,3}(x,y), d_{2,3}(x,y), and e_{2,3}(x,y) are homogeneous functions of x_0, x_1, y_0, y_1 of bidegrees (2,3) in the x_i and y_j<\/span>Consider now the subvariety S defined by the equations s_0 = s_1 = 0 in F.\u00a0 This is a copy of \\PP^1_{x_0,x_1} \\times \\PP^1_{y_0,y_1}; note that t=1 on S.\u00a0 The variety X meets S in the curve \\Gamma cut out by the equation a_1(y) = 0 inside S.\u00a0 Without loss of generality we can take a_1(y) = y_0, so that on \\Gamma we have y_0 = 0 and y_1 = 1.\u00a0 The curve \\Gamma is a copy of \\PP^1.\u00a0 We have that C is trivial on \\Gamma (because the section t of C is non-vanishing on \\Gamma) and that B is also trivial on \\Gamma (because the section y_1 of B is non-vanishing on \\Gamma).\u00a0 Thus -K_X = C-B is trivial on \\Gamma, and so X is not Fano. <\/span>
    \n<\/span><\/li>\n

  3. a divisor on \\PP^1 \\times \\PP^1 \\times \\PP^2 of tridegree (1,1,2).\u00a0 This is straightforward quantum Lefschetz:
    \nI_X(t) = \\sum_{k,l,m \\geq 0} t^{k+l+m} {(k+l+2m)! \\over (k!)^2 (l!)^2 (m!)^3}
    \nRegularizing this gives period sequence 31:
    \n1+20 t^2+132 t^3+1812 t^4+21720 t^5 + \\cdots<\/li>\n
  4. the blow-up of Y, the 2-to-1 cover of \\PP^1 \\times \\PP^2 with branch locus a divisor of type (2,2), with center a smooth fiber of Y \\to \\PP^1 \\times \\PP^2 \\to \\PP^2.\u00a0 The variety Y is a hypersurface of type 2L+2M in the rank-2 toric variety with weight data:
    \n\\begin{array}{ccccccc} s_0 & s_1 & x_0 & x_1 & x_2 & y & \\\\ 1 & 1 & 0 & 0 & 0 & 1 & L \\\\ 0& 0 & 1 & 1 & 1 & 1 & M \\end{array}
    \nWe need to blow up the locus x_1 = x_2 = 0, obtaining our variety X as a hypersurface of type 2L+2M in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & x_0 & t_1 & t_2 & y & x & \\\\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & L \\\\ 0& 0 & 1 & 0 & 0 & 1 & 1& M \\\\ 0 & 0 & 0 & 1 & 1 & 0 & -1 & N \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(2l+2m)! \\over l! l! m! n!n!(l+m)!(m-n)!}
    \nand regularizing gives period sequence 151:
    \n1+24 t^2+156 t^3+2280 t^4+27960 t^5 + \\cdots<\/li>\n
  5. the 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.\u00a0 We construct this as a codimension-2 complete\u00a0 intersection in F = \\PP(\\cO\\oplus\\cO\\oplus\\cO(-1,-1) \\to \\PP^1 \\times \\PP^2.\u00a0 F has weight data:
    \n\\begin{array}{ccccccccc} t_0 & t_1 & y_0 & y_1 & y_2 & x_0 & x_1 & x & \\\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & L \\\\ 0& 0 & 1 & 1 & 1 & 0 & 0 & -1 & M\\\\ 0 & 0 & 0 & 0& 0 &1 & 1 & 1 & N \\end{array}
    \nThe equations defining X are
    \n\\begin{pmatrix} y_0 & y_1 & t_0 A_2(y) \\\\ y_1 & y_2 & t_1 B_2(y) \\end{pmatrix} \\cdot \\begin{pmatrix} x_0 \\\\ x_1\\\\ x \\end{pmatrix} = 0
    \nwhere A_2, B_2 are quadratic polynomials in the y_i, and so X is a complete intersection of type (M+N)\\cdot(M+N) in F.\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+n} {(m+n)! (m+n)! \\over l! l!m! m!m!n!n!(n-m-l)!}
    \nand regularizing gives a period sequence that we do not have yet:
    \n1+32 t^2+204 t^3+3348 t^4+41040 t^5 + \\cdots<\/li>\n
  6. the blow-up of \\PP^3 with center the disjoint union of a line L and an elliptic curve \\Gamma of degree 4.\u00a0 Since \\Gamma is a (2,2) complete intersection in \\PP^2, our variety X is a hypersurface of type 2L+N\u00a0 in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & x & x_2 & x_3 & t_0 & t_1 & \\\\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & L\\\\1 & 1 & -1 & 0 & 0 & 0 & 0 & M\\\\0 & 0 & 0 & 0 & 0 & 1 & 1 & N \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(2l+n)! \\over m! m! (l-m)!l!l!n!n!}
    \nand regularizing gives period sequence 146:
    \n1+14 t^2+66 t^3+762 t^4+6960 t^5+ \\cdots<\/li>\n
  7. the blow-up X of W with center an elliptic curve which is a complete intersection of two members of |-{1 \\over 2} K_W|.\u00a0 Recall that W is a (1,1) hypersurface in \\PP^2 \\times \\PP^2, and so X is a complete intersection in \\PP^1 \\times \\PP^2 \\times\\PP^2 of type (0,1,1)\\cdot(1,1,1).\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(m+n)! (l+m+n)! \\over l! l!m! m!m!n!n!n!}
    \nand regularizing gives period sequence 36:
    \n1+10 t^2+48 t^3+438 t^4+3720 t^5+ \\cdots<\/p>\n

    Note that this is a D4 form (i.e. the Picard–Fuchs equation has unexpectedly low degree in D).\u00a0 It is almost certainly a G-Fano, as there is an obvious \\ZZ\/2\\ZZ-action.<\/li>\n

  8. a 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.\u00a0 The weight data for \\mathbb{F}_1 \\times \\PP^2 are:
    \n\\begin{array}{cccccccc} x_0 & s_1 & s_2 & x & y_0 & y_1 & y_2 & \\\\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & L\\\\0 & 1 & 1 &- 1 & 0 & 0 & 0 & M\\\\0 & 0 & 0 & 0 & 1 & 1 & 1 & N \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(l+2n)! \\over l! m! m!(l-m)!n!n!n!}
    \nand regularizing gives period sequence 85:
    \n1+12 t^2+54 t^3+540 t^4+4620 t^5 + \\cdots
    \n\\mathbb{F}_1 \\times \\PP^2 admits an obvious map to \\PP^1_{s_1,s_2} \\times \\PP^2_{y_0,y_1,y_2}.\u00a0 Writing the equation defining X in the form x_0 a + x b = 0 we find that a is a divisor of type (0,2) and b is a divisor of type (1,2).\u00a0 Thus X is also the blow up of \\PP^1 \\times \\PP^2 in a curve of bidegree (4,2) that is a complete intersection of type (0,2)\\cdot(1,2).<\/li>\n
  9. the blow-up of the cone W_4 \\subset \\PP^6 over the Veronese surface R_4 \\subset \\PP^5 with center the disjoint union of a vertex and a quartic in \\PP^2 \\cong R_4.\u00a0 Thus X is a hypersurface of type 4L+N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & s_2 & x & y & s & t & \\\\ 1 & 1 & 1 & 0 & -2 & 4 & 0 & L\\\\0 & 0 & 0 & 1 & 1 & -1 & 0 & M\\\\0 & 0 & 0 & 0 & 0 & 1 & 1 & N \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(4l+n)! \\over l! l! l!m!(-2l+m)!(4l-n+m)!n!}
    \nand regularizing gives period sequence 68:
    \n1+2 t^2+36 t^3+198 t^4+840 t^5+ \\cdots<\/li>\n
  10. the blow-up of a quadric Q in \\PP^4 with center a disjoint union of two conics on Q.\u00a0 We take Q to be the locus x_0 x_1 + x_2 x_3 + x_4^2 = 0 in \\PP^4_{x_0,x_1,x_2,x_3,x_4}, and take the conics to be cut out of Q by x_0 = x_1 = 0 and x_2 = x_3 = 0; note that the intersection of these two planes misses Q.\u00a0 So we can construct the variety X as a hypersurface of type 2L in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & t_2 & t_3 & x_4 & x & y & \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & L\\\\1 & 1 & 0 & 0 & 0 & -1 & 0 & M\\\\0 & 0 & 1 & 1 & 0 & 0 & -1 & N \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(2l)! \\over m!m! n!n!l!(l-m)!(l-n)!}
    \nand regularizing gives period sequence 67:
    \n1+10 t^2+36 t^3+366 t^4+2640 t^5+ \\cdots<\/li>\n
  11. the blow-up X of V_7 with center a complete intersection of two general members of {-1\/2} K_{V_7}.\u00a0 X is also the blow up of \\PP^1 \\times \\PP^2 in a curve of bidegree (2,3) that is a complete intersection of type (1,1)\\cdot(1,2).\u00a0 Consider the toric variety with weight data:
    \n\\begin{array}{cccccccc} t_0 & t_1 & x_0 & s_1 & s_2 & s_3 & x & \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & L\\\\0 & 0 & 0 & 1 & 1 & 1 & -1 & M\\\\1 & 1 & 0 & 0 & 0 & 0 & 0 & N \\end{array}
    \nThis is \\PP^1 \\times V_7, and X is cut out here as a hypersurface of type L+M+N.\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(l+m+n)! \\over n! n! l!m!m!m!(l-m)!}
    \nand regularizing gives period sequence 107:
    \n1+6 t^2+30 t^3+186 t^4+1380 t^5+ \\cdots<\/p>\n

    To see that X is a blow-up of \\PP^1 \\times \\PP^2 as claimed, note that \\PP^1 \\times V_7 admits an obvious map to \\PP^1_{t_0,t_1} \\times \\PP^2_{s_0,s_1,s_2}.\u00a0 Rewriting the equation defining X in the form x_0 a + x b = 0 we see that a is a divisor of type (1,1) and b is a divisor of type (1,2).\u00a0 This suffices.<\/li>\n

  12. the blow-up of \\PP^3 with centre a disjoint union of a twisted cubic and a line. As we will see, X is also the blow up of \\PP^1 \\times \\PP^2 in a curve of bidegree (3,2) that projects isomorphically to a conic in \\PP^2. We begin by exhibiting X as a complete intersection in a toric variety F. The twisted cubic \\Gamma is cut out of \\PP^3_{x_0,\\dots, x_3} by the equations:
    \n\\rk \\begin{pmatrix} x_0 & x_1 &x_ 2\\\\ x_1 & x_2 & x_3 \\end{pmatrix} <2
    \nThe blow up of \\PP^3 along \\Gamma is cut out of \\PP^3_{x_0,\\dots, x_3}\\times \\PP^2_{y_0,y_1,y_2} by the equation:
    \n\\begin{pmatrix} x_0 & x_1 & x_2 \\\\ x_1 & x_2 & x_3 \\end{pmatrix} \\cdot \\begin{pmatrix} y_0 \\\\ y_1\\\\y_2 \\end{pmatrix} = 0<\/p>\n

    Observe that \\Gamma is disjoint from the line L = \\{x_0=x_3=0\\}. We therefore blow up \\PP^3_{x_0,\\dots, x_3}\\times \\PP^2_{y_0,y_1,y_2} along the locus x_0=x_3=0, obtaining the toric variety F with weight data:
    \n\\begin{array}{ccccccccc} s_0 & x_1 & x_2 & s_3 & x & y_0 & y_1 & y_2 & \\\\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & L \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & M\\\\ 1& 0 & 0 & 1 & -1 & 0 & 0 & 0 & N \\end{array}
    \nThe equations defining X inside F are:
    \n\\begin{pmatrix} s_0 x & x_1 & x_2 \\\\ x_1 & x_2 & s_3 x \\end{pmatrix} \\cdot \\begin{pmatrix} y_0 \\\\ y_1\\\\y_2 \\end{pmatrix} = 0
    \nand so X is a complete intersection of type (L+M)\\cdot(L+M).\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(l+m)!(l+m)! \\over n! l!l!n!(l-n)!m!m!m!}
    \nand regularizing gives period sequence 144:
    \n1+8 t^2+30 t^3+240 t^4+1740 t^5+ \\cdots<\/p>\n

    To see that X is a blow-up of \\PP^1 \\times \\PP^2 as claimed, note that the map F \\to \\PP^1 \\times \\PP^2 is [x_0:x_1:x_2:s_3:x:y_0:y_1:y_2] \\mapsto [s_0:s_3:y_0:y_1:y_2].\u00a0 Rewriting the equations of X as:<\/p>\n

    \\begin{pmatrix} s_0 y_0 & y_1 & y_2 \\\\ s_3 y_2 & y_0 & y_1 \\end{pmatrix} \\cdot \\begin{pmatrix} x \\\\ x_1\\\\x_2 \\end{pmatrix} = 0
    \nwe see that X is the blow-up of \\PP^1 \\times \\PP^2 along the curve C given by:
    \n\\rk \\begin{pmatrix} s_0 y_0 & y_1 & y_2 \\\\ s_3 y_2 & y_0 & y_1 \\end{pmatrix} < 2
    \nThe equations defining C are:
    \n\\begin{cases} y_1^2 - y_0 y_2 = 0 \\\\ s_0 y_0^2 - s_3 y_1 y_2 = 0 \\\\ s_0 y_0 y_1 - s_3 y_2^2 = 0\\end{cases}
    \nand so C lies entirely within the “cylinder surface” y_1^2 - y_0 y_2 = 0.\u00a0 This cylinder surface is abstractly isomorphic to \\PP^1_{s_0,s_3} \\times \\PP^1_{t_0,t_1}, where y_0 = t_0^2, y_1 = t_0 t_1, y_2 = t_1^2.\u00a0 The equations of C become:
    \n\\begin{cases} t_0(s_0 t_0^3 - s_3 t_1^3) = 0 \\\\ t_1(s_0 t_0^3 - s_3 t_1^3) = 0\\end{cases}
    \nThus C is as described above.<\/li>\n

  13. the blow-up…<\/li>\n
  14. the blow-up of \\PP^3 with center the union of a cubic in a plane \\Pi and a point P not in \\Pi.\u00a0 Let \\Pi be x_0 = 0 in \\PP^3_{x_0,x_1,x_2,x_3}, and let P be x_1 = x_2 = x_3 = 0.\u00a0 Thus X is the blow-up of the curve \\Gamma \\subset V_7 given by:
    \n\\begin{cases} x_0 = 0 \\\\ a_3(s_1,s_2,s_3) = 0 \\end{cases}
    \nwhere the blow-up V_7 \\to \\PP^3 is [x_0:s_1:s_2:s_3:x] \\mapsto [x_0:s_1 x: s_2 x: s_3 x].\u00a0 Thus X is a hypersurface of type 3M+N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} x_0 & s_1 & s_2 & s_3 & x & s & t & \\\\ 1 & 0 & 0 & 0 & 1 & -1 & 0 & L \\\\ 0 & 1 & 1 & 1 & -1 & 3 & 0 & M\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+2m+n} {(3m+n)! \\over l!m!m!m!(l-m)!(3m+n-l)!n!}
    \nand regularizing gives period sequence 148:
    \n1+2 t^2+18 t^3+102 t^4+420 t^5+ \\cdots<\/li>\n
  15. the 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.\u00a0 X is the blow-up of \\PP^1 \\times \\PP^2 along a curve of bidegree (2,2) that is a complete intersection of type (0,2)\\cdot(1,1).\u00a0 To see this, we first exhibit X as a hypersurface in a toric variety.\u00a0 Note that the blow-up of \\PP^4_{x_0,x_1,x_2,x_3,x_4} along the plane x_3 = x_4 = 0 and the line x_0 = x_1 = x_2 = 0 is toric with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & s_2 & t_3 & t_4 & x & y & \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & L \\\\ 1 & 1 & 1 & 0 & 0 & -1 & 0 & M\\\\ 0 & 0 & 0 & 1 & 1 & 0 & -1 & N\\end{array}
    \nThe map to \\PP^4 here sends [s_0 : s_1 : s_2 : t_3 : t_4 : x : y] \\mapsto [s_0 x : s_1 x : s_2 x : t_3 y : t_4 y]; there is also a map to \\PP^1 \\times \\PP^2 given by [t_0 : t_1 : s_0 : s_1 : s_2].\u00a0 X is cut out of the above toric variety by a section of L+M.\u00a0 Thus we have:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(l+m)! \\over m!m!m!n!n!(l-m)!(l-n)!}
    \nand regularizing gives period sequence 112:
    \n1+6 t^2+18 t^3+138 t^4+780 t^5+ \\cdots
    \nTo see that X is the blow-up of \\PP^1 \\times \\PP^2 as claimed, write the equation defining X as x a + y b = 0.\u00a0 Then by homogeneity a is a section of (0,2) and b is a section of (1,1).<\/li>\n
  16. the blow-up of B_7 with center the strict transform of a twisted cubic through the blown-up point P \\in \\PP^3.\u00a0 Let P be the point x_1 = x_2 = x_3 = 0 in \\PP^3_{x_0,x_1,x_2,x_3}, and let \\Gamma be the curve given by
    \n\\rk \\begin{pmatrix} x_0 & x_1 & x_2 \\\\ x_1 & x_2 & x_3 \\end{pmatrix} < 2
    \nLet the blow-up B_7 \\to \\PP^3 be given by [x_0 : s_ 1 : s_2 : s_3 : x] \\mapsto [x_0 : s_1 x: s_2 x: s_3 x].\u00a0 Then the strict transform of \\Gamma in B_7 is given by:
    \n\\rk \\begin{pmatrix} x_0 & s_1 & s_2 \\\\ x s_1 & s_2 & s_3 \\end{pmatrix} < 2
    \nAs before, we introduce new variables y_0, y_1, y_2 and the toric variety F with weight data:
    \n\\begin{array}{ccccccccc} x_0 & s_1 & s_2 & s_3 & x & y_0 & y_1 & y_2 & \\\\ 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & L \\\\ 0 & 1 & 1 & 1 & -1 & 0 & -1 & -1 & M \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & N \\end{array}
    \nX is cut out by the equations:
    \n\\begin{cases} x_0 y_0 + s_1 y_1 + s_2 y_2 = 0 \\\\ s_1 x y_0 + s_2 y_1 + s_3 y_2 = 0\\end{cases}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+n} {(l+n)!(l+n)! \\over l!m!m!m!(l-m)!n!(l-m+n)!(l-m+n)!}
    \nand regularizing gives period sequence 119:
    \n1 + 4 t^2 + 18 t^3 + 84 t^4 + 540 t^5 + \\cdots<\/li>\n
  17. a smooth divisor on \\PP^1 \\times \\PP^1 \\times \\PP^2 of tridegree (1,1,1).\u00a0 This is straightforward quantum Lefschetz:
    \nI_X(t) = \\sum_{k,l,m \\geq 0} t^{k+l+2m} {(k+l+m)! \\over (k!)^2 (l!)^2 (m!)^3}
    \nRegularizing this gives period sequence 37:
    \n1 + 4 t^2 + 12 t^3 + 84 t^4 + 360 t^5 + \\cdots
    \nNote that X is also the blow-up of \\PP^1 \\times \\PP^2 along a curve of bidegree (1,2).<\/li>\n
  18. the blow-up of \\PP^3 with center a disjoint union of a line L and a conic.\u00a0 Take the line to be \\{x_0 = x_1 = 0\\} \\subset \\PP^3_{x_0,x_1,x_2,x_3}, and take the conic to be x_0 x_1 + x_2^2 = x_3 = 0.\u00a0 The blow-up of \\PP^3 in L is [s_0 : s_1 : x : x_2 : x_3] \\mapsto [s_0 x : s_1 x : x_2 : x_3], and the strict transform of the conic is cut out by x_3 = s_0 s_1 x^2 + x_2^2 = 0.\u00a0 Thus X is a hypersurface of type 2L+N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & x & x_2 & x_3 & s & t & \\\\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & L \\\\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & M \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N \\end{array}
    \ncut out by the equation s x_3 = t(x_2^2+s_0 s_1 x^2). Quantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+m+n} {(2l+n)! \\over m!m!(l-m)l!l!(l+n)!n!}
    \nand regularizing gives period sequence 160:
    \n1+4 t^2+18 t^3+60 t^4+480 t^5+ \\cdots<\/li>\n
  19. the blow-up of a quadric Q with center two non-colinear points.\u00a0 We construct this by taking the equation of the quadric to be x_0 x_1 + x_2 x_3 + x_4^2 = 0 inside \\PP^4_{x_0,x_1,x_2,x_3,x_4}, blowing up the line x_2 = x_3 = x_4 = 0 in \\PP^4 (this line is not contained in Q) and then taking the proper transform of the quadric inside the blow-up of \\PP^4.\u00a0 This is a hypersurface of type 2L in the toric variety with weight data
    \n\\begin{array}{ccccccc} x_0 & x_1 & s_2 & s_3 & s_4 & x & \\\\ 1 & 1 & 0 & 0 & 0 & 1 & L \\\\ 0 & 0 & 1 & 1 & 1 & -1 & M \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m \\geq 0} t^{l+2m} {(2l)! \\over l!l!m!m!m!(l-m)!}
    \nand regularizing gives period sequence 57:
    \n1+2 t^2+12 t^3+54 t^4+240 t^5+ \\cdots<\/li>\n
  20. the blow-up of a 3-dimensional quadric Q with center two disjoint lines on it.\u00a0 We take the quadric with equation x_0 x_3 + x_1 x_2 + x_4^2 in \\PP^4_{x_0,x_1,x_2,x_3,x_4} and blow up the lines x_0=x_1=x_4=0 and x_2=x_3=x_4=0.\u00a0 Thus X is a hypersurface of type M+N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} s_0 & s_1 & s_2 & s_3 & s & x & y & \\\\ 0 & 0 & 0 & 0 & -1 & 1 & 1 & L \\\\ 1 & 1 & 0 & 0 & 1 & -1 & 0 & M \\\\ 0 & 0 & 1 & 1 & 1 & 0 & -1 & N\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m \\geq 0} t^{l+m+n} {(m+n)! \\over m!m!n!n!(-l+m+n)!(l-m)!(l-n)!}
    \nand regularizing gives period sequence 63:
    \n1+4 t^2+12 t^3+60 t^4+360 t^5+ \\cdots<\/li>\n
  21. the blow-up of \\PP^1 \\times \\PP^2 along a curve of bidegree (2,1).\u00a0\u00a0 X is described by a single equation sy_0 +t (x_0 q_0 + x_1 q_1)=0, where q_0, q_1 are quadratic polynomials in y_1, y_2, in the toric variety with weight data
    \n\\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & y_2 & s & t& \\\\ 1 & 1 & 0 & 0 & 0 & 0 & -1 & L \\\\ 0 & 0 & 1 & 1 & 1 & 0 & -1 & M \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(m+n)! \\over l!l!m!m!m!n!(n-l-m)!}
    \nand regularizing gives period sequence 98:
    \n1+6 t^2+6 t^3+114 t^4+240 t^5+ \\cdots<\/li>\n
  22. the blow-up of \\PP^1 \\times \\PP^2 along a curve of bidegree (0,2), that is a conic in \\{t\\}\\times \\PP^2. Thus X is described by a single equation sx_0 +t (y_0y_2-y_1^2)=0 in the toric variety with weight data
    \n\\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & y_2 & s & t& \\\\ 1 & 1 & 0 & 0 & 0 & -1 & 0 & L \\\\ 0 & 0 & 1 & 1 & 1 & 0 & -2 & M \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {n! \\over l!l!m!m!m!(n-l)!(n-2m)!}
    \nand regularizing gives period sequence 152:
    \n1+2 t^2+6 t^3+54 t^4+180 t^5+ \\cdots<\/li>\n
  23. the blow-up of B_7 with center a conic passing through the blown-up point P \\in \\PP^3. Let P \\in \\PP^3_{x_0,x_1,x_2,x_3} be the point x_1 = x_2 = x_3 =0.\u00a0 Define the conic by:
    \n\\begin{cases} x_3 = 0 \\\\ x_0 x_1 + x_2^2 = 0\\end{cases}
    \nTaking the proper transform under the blow-up B_7 \\to \\PP^3 gives the equations:
    \n\\begin{cases} s_3 = 0 \\\\ x_0 s_1 + x s_2^2 = 0\\end{cases}
    \nwhere the blow-up is [x_0 : s_1 : s_2 : s_3 : x] \\mapsto [x_0 : x s_1 : x s_2 : x s_3], and so we need to consider the locus:
    \ns(s_3) + t(x_0 s_1 + x s_2^2) = 0
    \nThis is a hypersurface of type L+M+N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} x_0 & s_1 & s_2 & s_3 & x & s & t & \\\\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & L \\\\ 0 & 1 & 1 & 1 & -1 & 0 & 0 & M\\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+m+n} {(l+m+n)! \\over l!m!m!m!(l-m)!(l+n)!n!}
    \nand regularizing gives period sequence 158:
    \n1+2 t^2+12 t^3+30 t^4+180 t^5+\\cdots<\/li>\n
  24. the fiber product X = W \\times_{\\PP^2} \\mathbb{F}_1 where W is a (1,1) hypersurface in \\PP^2 \\times \\PP^2.\u00a0 This is the blow up of \\PP^1 \\times \\PP^2 along a curve of bidegree (1,1).\u00a0 To see this, note first that X is cut out of \\PP^2_{x_0,x_1,x_2} \\times \\PP^2_{y_0,y_1,y_2} \\times \\PP^1_{s_0,s_1} by the equations:
    \n\\begin{cases} y_0 x_0 + y_1 x_1 + y_2 x_2 = 0 \\\\ s_0 x_0 + s_1 x_1 = 0 \\end{cases}
    \nThe first equation here cuts W out of \\PP^2_{x_0,x_1,x_2} \\times \\PP^2_{y_0,y_1,y_2}; the second equation cuts \\mathbb{F}_1 out of \\PP^2_{y_0,y_1,y_2} \\times \\PP^1_{s_0,s_1}, as it is the equation defining the blow-up of \\PP^2.<\/p>\n

    We now exhibit X as the blow-up of a curve in \\PP^1 \\times \\PP^2 by projecting to \\PP^2_{y_0,y_1,y_2} \\times \\PP^1_{s_0,s_1}.\u00a0 This projection is an isomorphism away from the locus where the matrix
    \n\\begin{pmatrix} y_0 & y_1 & y_2 \\\\ s_0 & s_1 & 0 \\end{pmatrix}
    \ndrops rank.\u00a0 This locus is:
    \n\\begin{cases} y_2 = 0 \\\\ y_0 s_1 - y_1 s_0 = 0\\end{cases}
    \ni.e. a curve in \\PP^1 \\times \\PP^2 of bidegree (1,1).
    \nWe can further simplify things by writing X as a hypersurface in \\PP^2 \\times \\mathbb{F}_1.\u00a0 Write the co-ordinates on \\PP^2 as y_0, y_1, y_2 and the co-ordinates on \\mathbb{F}_1 as t_0, t_1, x, x_2; here the blow-up \\mathbb{F}_1 \\to \\PP^2 sends [t_0 : t_1 : x : x_2] \\mapsto [t_0 x : t_1 x : x_2].The two equations defining X (given above) reduce to the single equation:
    \nt_0 x y_0 + t_1 x y_1 + x_2 y_2 = 0
    \nThus X is a hypersurface of type L+N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} y_0 & y_1 & y_2 & s_0 & s_1 & x & x_2 & \\\\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & L \\\\ 0 & 0 & 0 & 1 & 1 & -1 & 0 & M \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+m+n} {(l+n)! \\over l!l!l!m!m!(n-m)!n!}
    \nand regularizing gives period sequence 86:
    \n1+4 t^2+6 t^3+60 t^4+180 t^5 + 1210 t^6 + 5460 t^7 + 30940 t^8 + 165480 t^9 +\\cdots<\/li>\n

  25. the blow-up of \\PP^3 with center two disjoint lines.\u00a0 This is the toric variety with weight data:
    \n\\begin{array}{ccccccc} s_0 & s_1 & t_0 & t_1 & x & y & \\\\ 0 & 0 & 0 & 0 & 1 & 1 & L \\\\ 1 & 1 & 0 & 0 & -1 & 0 & M\\\\ 0 & 0 & 1 & 1 & 0 & -1 & N\\end{array}
    \nThe blow-up map is [s_0 : s_1 : t_0 : t_1 : x : y] \\mapsto [s_0 x : s_1 x : t_0 y : t_1 y ].\u00a0 We have:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+m+n} {1 \\over m!m!n!n!(l-m)!(l-n)!}
    \nand regularizing gives period sequence 41:
    \n1+2 t^2+12 t^3+30 t^4+120 t^5 + \\cdots<\/li>\n
  26. the blow-up X of \\PP^3 with center a disjoint union of a point and a line.\u00a0 This is also the blow-up of \\PP^1 \\times \\PP^2 in a curve of bidegree (0,1).\u00a0 So X is a toric variety, with weight data:
    \n\\begin{array}{ccccccc} s_0 & x_1 & t_0 & y_1 & y_2 & x & \\\\ 1 & 1 & 0 & 0 & 0 & 0 & L \\\\ 0 & 0 & 1 & 1 & 1 & 0 & M\\\\ -1 & 0 & -1 & 0 & 0 & 1 & N\\end{array}
    \nThe blow-up map is [s_0 : x : t_0 : y_1 : y_2 : x] \\mapsto [s_0 x : x_1 : t_0 x : y_1 : y_2].\u00a0 We have:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+3m-n} {1 \\over (l-n)!l!(m-n)!m!m!n!}
    \nand regularizing gives period sequence 113:
    \n1+2 t^2+6 t^3+30 t^4+120 t^5 + \\cdots<\/li>\n
  27. X = \\PP^1 \\times \\PP^1 \\times \\PP^1.\u00a0 This gives:
    \nI_X = \\sum_{k,l,m \\geq 0} t^{2k+2l+2m} {1 \\over k!k!l!l!m!m!}
    \nand regularizing gives period sequence 21:
    \n1 + 6 t^2 + 90 t^4 + 0 t^5 + \\cdots<\/li>\n
  28. X = \\PP^1 \\times \\mathbb{F}_1.\u00a0 This gives:
    \nI_X = \\sum_{k,l,m \\geq 0} t^{2k+l+2m} {1 \\over k!k!l!l!(m-l)!m!}
    \nand regularizing gives period sequence 90:
    \n1+4 t^2+6 t^3+36 t^4+180 t^5 + \\cdots<\/li>\n
  29. the blow-up of B_7 with center a line on the exceptional divisor of the blow-up B_7 \\to \\PP^3.\u00a0 This is a toric variety with weight data:
    \n\\begin{array}{ccccccc} x_0 & s_1 & s_2 & s & x & y & \\\\ 1 & 0 & 0 & 0 & 1 & 0 & L \\\\ 0 & 1 & 1 & 1 & -1 & 0 & M\\\\ 0 & 0 & 0 & 1 & 1 & -1 & N\\end{array}
    \nWe have:
    \nI_X = \\sum_{l,m,n \\in \\ZZ} t^{2l+2m+n} {1 \\over l!m!m!(m+n)!(l-m+n)!(-n)!}
    \nand regularizing gives period sequence 163:
    \n1+2 t^2+30 t^4+60 t^5+ \\cdots<\/li>\n
  30. the blow-up of B_7 with center the strict transform of a line through the blown-up point P \\in \\PP^3.\u00a0 This is a toric variety with weight data:
    \n\\begin{array}{ccccccc} x_0 & s_1 & t_2 & t_3 & x & y & \\\\ 1 & 0 & 0 & 0 & 1 & 0 & L \\\\ 0 & 1 & 0 & 0 & -1 & 1 & M\\\\ 0 & 0 & 1 & 1 & 0 & -1 & N\\end{array}
    \nWe have:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{2l+m+n} {1 \\over l!m!n!n!(l-m)!(m-n)!}
    \nand regularizing gives period sequence 84:
    \n1+2 t^2+6 t^3+30 t^4+60 t^5+ \\cdots<\/li>\n
  31. the total space of the bundle \\PP (\\cO \\oplus \\cO(1,1)) over \\PP^1 \\times \\PP^1. This is the toric variety with weight data:
    \n\\begin{array}{ccccccc} 1 & 1 & 0 & 0 & 0 & 1 & L \\\\ 0 & 0 & 1 & 1 & 0 & 1 & M\\\\ 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nWe have:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{3l+3m+2n} {1 \\over l!l!m!m!n!(l+m+n)!}
    \nand regularizing gives period sequence 53:
    \n1 + 2 t^2 + 12 t^3 + 6 t^4 + 120 t^5+ \\cdots<\/li>\n<\/ol>\n

    Note<\/h2>\n

    Consider the hypersurface of type 2N in the toric variety with weight data:
    \n\\begin{array}{cccccccc} 1 & 1 & 0 & 0 & 0 & 0 & -1 & L \\\\ 0 & 0 & 1 & 1 & 0 & 0 & -1 & M\\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & N\\end{array}
    \nWe have:
    \nI_X = \\sum_{l,m,n \\geq 0} t^{l+m+n} {(2n)! \\over l!l!m!m!n!n!(-l-m+n)!}
    \nand regularizing gives period sequence 32:
    \n1 + 10 t^2 + 24 t^3 + 318 t^4 + 1680 t^5+ \\cdots
    \nIt seems that Mori-Mukai may have missed this variety, and have included number 2 in the rank 3 list by mistake.\u00a0 Note that our X is a section of 2 P where P is the tautological bundle on \\PP_{\\PP^1 \\times \\PP^1}(\\cO \\oplus \\cO \\oplus \\cO(-1,-1)).\u00a0 The degree of X is 28.
    \n<\/span>Never mind: this is #2 on the Mori-Mukai list of rank-4 Fanos.<\/p>\n","protected":false},"excerpt":{"rendered":"

    a double cover of branched along a divisor of tridegree (2,2,2).\u00a0 This is a hypersurface of type in the toric variety with weight data: Quantum Lefschetz gives: and regularizing gives period sequence 22: Note that this is a G-Fano. a member of on the -bundle over such that is irreducible, where is the tautological line […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":277,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-228","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/228","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=228"}],"version-history":[{"count":72,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/228\/revisions"}],"predecessor-version":[{"id":236,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/228\/revisions\/236"}],"up":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/277"}],"wp:attachment":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}