{"id":268,"date":"2010-09-07T12:38:58","date_gmt":"2010-09-07T12:38:58","guid":{"rendered":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=268"},"modified":"2012-02-27T12:56:55","modified_gmt":"2012-02-27T12:56:55","slug":"higher-rank-fanos","status":"publish","type":"page","link":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/?page_id=268","title":{"rendered":"Higher rank Fano 3-folds"},"content":{"rendered":"

Rank 4 Fanos<\/h1>\n
    \n
  1. This is divisor of degree (1,1,1,1) on \\PP^1 \\times \\PP^1 \\times \\PP^1 \\times \\PP^1
    \nI_X = \\sum_{a,b,c,d \\geq 0} \\frac{(a+b+c+d)!}{a!^2b!^2c!^2d!^2} t^{a+b+c+d}
    \nRegularizing gives period sequence 3:
    \n1+12 t^2+48 t^3+540 t^4+4320 t^5 + \\cdots<\/li>\n
  2. the blow-up of the cone over a smooth quadric surface S in \\PP^3 with center the disjoint union of the vertex and an elliptic curve on S.\u00a0 The blow-up of the cone over S with center the vertex is the toric variety with weight data:
    \n\\begin{array}{ccccccc} s_0 & s_1 & t_0 & t_1 & x & y & \\\\ 1 & 1 & 0 & 0 & 0 & -1 & \\\\ 0 & 0 & 1 & 1 & 0 & -1 & \\\\ 0 & 0 & 0 & 0 & 1 & 1 & \\end{array}
    \nThe morphism to \\PP^4 is given by [s_0 : s_1 : t_0 : t_1 : x : y] \\mapsto [x : y s_0 t_0 : y s_1 t_1 : y s_0 t_1 : y s_1 t_0]; the image here is x_1 x_2 - x_3 x_4 = 0 in \\PP^4_{x_0,x_1,x_2,x_3,x_4}. To obtain X, we blow up the elliptic curve x = f_{2,2}(s,t) = 0.\u00a0 Thus X is the hypersurface u x + v f_{2,2}(s,t) = 0 in the toric variety with weight data:
    \n\\begin{array}{ccccccccc} s_0 & s_1 & t_0 & t_1 & x & y & u& v&\\\\ 1 & 1 & 0 & 0 & 0 & -1 & 2 & 0 & A \\\\ 0 & 0 & 1 & 1 & 0 & -1 & 2 & 0 & B\\\\ 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & C \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & D \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{a,b,c,d \\geq 0} \\frac{(2a+2b+d)!}{a! a!b!b!b! c! (c-b-a)! (2a+2b-c+d)! d!} t^{a+b+c+d}
    \nand regularizing gives period sequence 32:
    \n1+10 t^2+24 t^3+318 t^4+1680 t^5 + \\cdots<\/li>\n
  3. The blow-up of \\PP^1 \\times \\PP^1 \\times \\PP^1 with center a curve \\Gamma of tridegree (1,1,2).\u00a0 We can take \\Gamma to be parametrized as [s_0:s_1] \\mapsto [s_0:s_1:s_0:s_1:s_0^2:s_1^2] \\subset \\PP^1_{x_0,x_1} \\times \\PP^1_{y_0,y_1} \\times \\PP^1_{z_0,z_1}.\u00a0 We embed \\PP^1_{x_0,x_1} \\times \\PP^1_{y_0,y_1} \\times \\PP^1_{z_0,z_1} into \\PP^3_{u_0,u_1,u_2,u_3} \\times \\PP^1_{z_0,z_1} via the map u_0 = x_0 y_0, u_1 = x_1 y_1, u_2 = x_0 y_1, u_3 = x_1 y_0.\u00a0 Now, in \\PP^3 \\times \\PP^1, \\Gamma becomes the complete intersection defined by equations:
    \n\\begin{cases} u_2 - u_3 = 0\\\\ u_0 u_1 - u_2 u_3 = 0 \\\\ u_0 z_1 - u_1 z_0 = 0 \\end{cases}
    \nNote that the second equation here is just the equation of X inside \\PP^2 \\times \\PP^1.\u00a0 Thus we can blow up \\PP^3 \\times \\PP^1 along the locus:
    \n\\begin{cases} u_2 - u_3 = 0 \\\\ u_0 z_1 - u_1 z_0 = 0 \\end{cases}
    \nand then construct X by imposing the strict transform of the remaining equation u_0 u_1 - u_2 u_3 = 0.\u00a0 This exhibits X as a complete intersection of type (L+M+N) \\cdot (2L) inside the toric variety with weight data:
    \n\\begin{array}{ccccccccc} u_0 & u_1 & u_2 & u_3 & z_0 & z_1 & s& t&\\\\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & L \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & M\\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\\end{array}
    \nNote that this is rank 4 even though the ambient space has rank 3; there is no contradiction here since the line bundle L is not ample and so Lefschetz fails.\u00a0 Quantum Lefschetz (which does not fail) gives:
    \nI_X = \\sum_{l,m,n \\geq 0} \\frac{(l+m+n)!(2l)!}{l!l!l!l!m!m!(m+n)!n!} t^{l+2m+n}
    \nand regularizing gives period sequence 122:
    \n1+8 t^2+24 t^3+216 t^4+1320 t^5+ \\cdotsA cleaner and more systematic development is as follows.\u00a0 The curve \\Gamma is defined scheme-theoretically by the equations:
    \n\\begin{cases} x_0 y_1 - x_1 y_0 = 0 \\\\ z_1 x_0 y_0 - z_0 x_1 y_1 = 0\\end{cases}
    \ninside \\PP^1_{x_0,x_1} \\times \\PP^1_{y_0,y_1} \\times \\PP^1_{z_0,z_1}.\u00a0 (Please look at the parametrization given above.)\u00a0 So X is given by the equation s(x_0 y_0 - x_1 y_1) - t (x_0 y_0 z_1 - x_1 y_1 z_0) = 0 inside the toric variety with weight data:
    \n\\begin{array}{ccccccccc} x_0 & x_1 & y_0 & y_1 & z_0 & z_1 & s& t&\\\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & A \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & B\\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & C \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & D\\end{array}
    \nNow Quantum Lefschetz gives:
    \nI_X = \\sum_{a, b, c, d \\geq 0} \\frac{(a+b+c+d)!}{a!a!b!b!c!c!(c+d)!d!} t^{a+b+2c+d}
    \nand regularizing gives period sequence 122:
    \n1+8 t^2+24 t^3+216 t^4+1320 t^5+ \\cdots<\/li>\n
  4. the blow-up of Y (rank 3, number 19; the blow-up of a quadric with center two non-colinear points P, Q) with center the strict transform of a conic containing P and Q.\u00a0 Consider the line x_2 = x_3 = x_4 = 0 inside \\PP_{x_0,x_1,x_2,x_3,x_4}^4, and also the plane x_3 = x_4 = 0.\u00a0 Let F \\to \\PP^4 be the blow-up of the line followed by the strict transform of the plane (in that order); this is the toric variety with weight data:
    \n\\begin{array}{ccccccccc} x_0 & x_1 & s_2 & t_3 & t_4 & x & s& \\\\ 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}
    \nThe variety X is the strict transform of a general quadric in \\PP^4; in other words it is a hypersurface of type 2L in F.\u00a0 (Note that X is rank 4 even though the ambient space is rank 3; there is no contradiction here because 2L is not ample.)\u00a0 Quantum Lefschetz gives:
    \nI_X = \\sum_{l, m, n \\geq 0} \\frac{(2l)!}{l!l!m!n!n!(l-m)!(m-n)!} t^{l+m+n}
    \nand regularizing gives period sequence 103:
    \n1+6 t^2+24 t^3+138 t^4+960 t^5+ \\cdots<\/li>\n
  5. The blow-up of \\PP^1 \\times \\PP^2 with center two disjoint curves, one of bidegree (2,1) and the other of bidegree (1,0).\u00a0 This is a hypersurface of type 2A+C+D in the toric variety with weight data:
    \n\\begin{array}{ccccccccc} s_0 & s_1 & x_2 & x & t_0 & t_1 & u& v&\\\\ 1 & 1 & 0 & -1 & 0 & 0 & 2 & 0 & A \\\\ 0 & 0 & 1 & 1 & 0 & 0 & -1 & 0 & B\\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & C \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & D\\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{a, b, c, d \\geq 0} \\frac{(2a+c+d)!}{a!a!b!(b-a)!c!c!(2a-b+c+d)!d!} t^{a+b+2c+d}
    \nand regularizing gives period sequence 147:
    \n1+8 t^2+18 t^3+192 t^4+960 t^5+ \\cdots<\/li>\n
  6. the blow-up of \\PP^1 \\times \\PP^1 \\times \\PP^1 with center the curve of tridegree (1,1,1).\u00a0 This curve is cut out of \\PP^1_{x_0,x_1} \\times \\PP^1_{y_0,y_1} \\times \\PP^1_{z_0,z_1} by the equations
    \n\\rk \\begin{pmatrix} x_0 & y_0 & z_0 \\\\ x_1 & y_1 & z_1 \\end{pmatrix} < 2
    \nThus the blow-up X is cut out of the toric\u00a0 variety with weight data:
    \n\\begin{array}{cccccccccc} x_0 & x_1 & y_0 & y_1 & z_0 & z_1 & u_0 & u_1 & u_2 & \\\\ 1 & 1 & 0 & 0 & 0 & 0 &0 & 1 & 1 & A \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 0 & B \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 & C \\\\ 0& 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & D \\end{array}
    \nby the equation:
    \n\\rk \\begin{pmatrix} x_0 & y_0 & z_0 \\\\ x_1 & y_1 & z_1 \\end{pmatrix} \\cdot \\begin{pmatrix} u_0 \\\\ u_1 \\\\ u_2 \\end{pmatrix} = 0
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{a,b,c,d \\geq 0} \\frac{(a+d)!(a+d)!}{a! a!b!b! c!c!d! (a-b+d)! (a-c+d)!} t^{2a+b+c+d}
    \nand regularizing gives period sequence 65:
    \n1+6 t^2+18 t^3+114 t^4+720 t^5 + \\cdots<\/li>\n
  7. The blow-up of W \\subset \\PP^2 \\times \\PP^2 (a divisor of type (1,1)) with center two disjoint curves on it, of bidegree (0,1) and (1,0).\u00a0 We define W as the zero locus of x_0 y_0 + x_1 y_1 + x_2 y_2 = 0 inside \\PP^2_{x_0,x_1,x_2} \\times \\PP^2_{y_0,y_1,y_2}.\u00a0 Blowing up the disjoint union of x_0 = x_1 = 0 and y_0 = y_1 = 0 in \\PP^2 \\times \\PP^2 induces the blow-up that we seek.\u00a0 Thus X is a hypersurface of type A+B in the toric variety with weight data:
    \n\\begin{array}{ccccccccc} s_0 & s_1 & x_2 & t_0 & t_1 & y_2 & u & v & \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & A \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & B \\\\ 1 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & C \\\\ 0& 0 & 0 & 1 & 1 & 0 & 0 & -1 & D \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{a,b,c,d \\geq 0} \\frac{(a+b)!}{c!c!a!d!d!b!(a-c)!(b-d)!} t^{a+b+c+d}
    \nand regularizing gives period sequence 69:
    \n1+6 t^2+12 t^3+114 t^4+480 t^5 + \\cdots<\/li>\n
  8. the blow-up of \\PP^1 \\times \\PP^1 \\times \\PP^1 with center a curve \\Gamma of tridegree (0,1,1).\u00a0 The curve \\Gamma is cut out of \\PP^1_{x_0,x_1} \\times \\PP^1_{y_0,y_1} \\times \\PP^1_{z_0,z_1} by the equations
    \n\\begin{cases} z_0 = 0 \\\\ x_0 y_0 + x_1 y_1 = 0 \\end{cases}
    \nand so X is cut out of the toric variety with weight data:
    \n\\begin{array}{ccccccccc} x_0 & x_1 & y_0 & y_1 & z_0 & z_1 & s & t & \\\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & A \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & B \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & C \\\\ 0& 0 & 0 & 0 & 0 & 0 & 1 & 1 & D \\end{array}
    \nQuantum Lefschetz gives:
    \nI_X = \\sum_{a,b,c,d \\geq 0} \\frac{(c+d)!}{a!a!b!b!c!c!d!(d+c-b-a)!} t^{a+b+2c+d}
    \nand regularizing gives period sequence 105:
    \n1+6 t^2+12 t^3+90 t^4+480 t^5 + \\cdots<\/li>\n
  9. This is the toric variety with weight data:
    \n\\begin{array}{cccccccc} 1 & 1 & 0 & 0 & -1 & 0 & 0 & A \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & B \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & C \\\\ 0& 0 & 0 & 1 & 1 & 0 & -1 & D \\end{array}
    \nIts regularized period sequence is period sequence 102:
    \n1+4 t^2+12 t^3+60 t^4+300 t^5 + \\cdots<\/li>\n
  10. This is the toric variety with weight data:
    \n\\begin{array}{cccccccc} 0 & 1 & -1 & 0 & 1 & 0 & 0 & A \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & B \\\\ 1 & 0 & 1 & 0 & -1 & 0 & 0 & C \\\\ 0& 0 & 0 & 0 & 0 & 1 & 1 & D \\end{array}
    \nIts regularized period sequence is period sequence 142:
    \n1+6 t^2+6 t^3+90 t^4+240 t^5 + \\cdots<\/li>\n
  11. This is the toric variety with weight data:
    \n\\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & -1 & 1 & A \\\\ 0 & 0 & 1 & 1 & 0 & -1 & 0 & B \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & C \\\\ 0& 1 & 0 & 0 & 0 & 1 & -1 & D \\end{array}
    \nIts regularized period sequence is period sequence 93:
    \n1+4 t^2+12 t^3+36 t^4+300 t^5 + \\cdots<\/li>\n
  12. This is the toric variety with weight data:
    \n\\begin{array}{cccccccc} 0 & 0 & 0 & 0 & -1 & 1 & 1 & A \\\\ 1 & 1 & 0 & 0 & -1 & 0 & 0 & B \\\\ 0 & 0 & 1 & 0 & 1 & -1 & 0 & C \\\\ 0& 0 & 0 & 1 & 1 & 0 & -1 & D \\end{array}
    \nIts regularized period sequence is period sequence 150:
    \n1+4 t^2+6 t^3+60 t^4+120 t^5 + \\cdots<\/li>\n
  13. (degree 26, see the erratum)
    \nThis is the blow-up of \\PP^1 \\times \\PP^1 \\times \\PP^1 along a curve \\Gamma of tridegree (1,1,3).\u00a0 Note that \\Gamma is a complete intersection of type (1,1,0)\\cdot(2,1,1), and thus we can realize X as a hypersurface of type A+B+D in the toric variety F with weight data:
    \n\\begin{array}{ccccccccc} 1& 1 & 0 & 0 & 0 & 0 & 0 &-1 & A \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & B \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & C \\\\ 0& 0 & 0 & 0 & 0 & 0 & 1 & 1 & D \\end{array}
    \nWe have -K_X = B+C+D.\u00a0 This is ample on X but only semi-positive on the ambient space F.\u00a0 Thus we are in the situation described here<\/a>, and we use the same notation.\u00a0 We have:
    \n\\begin{cases} F(q) = 1 \\\\ G(q) = q_2+q_4+2q_1 q_4 \\\\ H_1(q) = \\sum_{k>0} {(-1)^k \\over k} q_1^k = {-\\log(1+q_1)} \\\\ H_2(q) = 0 \\\\ H_3(q) = {-\\log(1+q_1)} \\\\ H_4(q) = \\log(1+q_1) \\end{cases}
    \nInverting the mirror map gives:
    \n\\begin{cases} q_1 = {\\hat{q}_1 \\over 1 - \\hat{q}_1} \\\\ q_2 = \\hat{q}_2 \\\\ q_3 = {\\hat{q}_3 \\over 1 - \\hat{q}_1} \\\\ q_4 = \\hat{q}_4(1-\\hat{q}_1) \\end{cases}
    \nThus the cohomological-degree-zero part of the J-function of X is:
    \n\\exp(-\\hat{q}_2-\\hat{q}_4(1-\\hat{q}_1)-2\\hat{q}_1\\hat{q}_4) \\sum_{a,b,c,d \\geq 0} \\hat{q}_1^a \\hat{q}_2^b \\hat{q}_3^c \\hat{q}_4^d (1-\\hat{q}_1)^{d-c-a} {(a+b+d)! \\over a!a!b!b!c!c!d!(d-c-a)!} z^{-b-c-d}
    \nWe construct the regularized period sequence from this by making the change of variables \\hat{q}_1 = 1, \\hat{q}_2 = t, \\hat{q}_3 = t, \\hat{q}_4 = t, z = 1:
    \n\\exp(-3t) \\sum_{a,b,c \\geq 0} t^{a+b+2c} {(2a+b+c)! \\over a!a!b!b!c!c!(a+c)!}
    \nand then doing the trick with factorials:
    \nI_{reg}(t) = 1+12 t^2+42 t^3+468 t^4+3360 t^5+31350 t^6 + \\cdots
    \nThis is period sequence 88.<\/li>\n<\/ol>\n

    Rank 5 Fanos<\/h1>\n
      \n
    1. the blowup of Y, where Y is number 29 on the rank-2 list (i.e. the blow-up of the quadric 3-fold Q with center a conic), with center 3 exceptional lines of the blow-up Y \\to Q.\u00a0 To compute this, take the defining equation of the quadric to be
      \nx_0 x_1 + x_1 x_2 + x_2 x_0 + x_3 x_4
      \nin \\PP^4_{x_0,x_1,x_2,x_3,x_4} and blow up the ambient space \\PP^4 in the plane \\Pi = \\{x_3=x_4=0\\} .\u00a0 Note that the conic in $\\Pi$ contains the co-ordinate points [1:0:0], [0:1:0], [0:0:1].\u00a0 Blowing up the exceptional lines over these points exhibits X as a hypersurface of type 2A+2B+C+D+E in the toric variety with weight data:
      \n\\begin{array}{cccccccccc} x_0 & x_1 & x_2 & s_3 & s_4 & x & t_{01} & t_{02} & t_{12} & \\\\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & A \\\\ 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & B \\\\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & C \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & D \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & E \\end{array}
      \nThe regularized period sequence is:
      \n1+10 t^2+42 t^3+342 t^4+2640 t^5+21250 t^6+180600 t^7 + \\cdots
      \nThis is period sequence 114.<\/li>\n
    2. This is the toric variety with weight data:
      \n\\begin{array}{cccccccccc} x_0 & x_1 & y_0 & y_1 & s & t & u & v & \\\\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & A \\\\ 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & B \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & C \\\\ 0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 & D \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & E \\end{array}
      \nThe regularized period sequence is:
      \n1+6 t^2+18 t^3+114 t^4+660 t^5+3930 t^6+25620 t^7+ \\cdots
      \nThis is period sequence 87.<\/li>\n
    3. \\PP^1 \\times S_6
      \nThe I-function is the product:
      \nI_{\\PP^1}(t) I_{S_6}(t) = 1+4 t^2+2 t^3+7 t^4+5 t^5+\\frac{265 t^6}{36}+\\frac{11 t^7}{2} + \\cdots
      \nand the regularized period sequence is:
      \n1+8 t^2+12 t^3+168 t^4+600 t^5+5300 t^6+27720 t^7+ \\cdots
      \nThis is period sequence 43.<\/li>\n<\/ol>\n

      Rank > 5 Fanos<\/h1>\n

      These are products \\PP^1 \\times S_d of line with del Pezzo surface of degree d \\leq 5.\u00a0 The I-series are products of I-series for line and for del Pezzo surfaces.<\/p>\n

        \n
      1. \\PP^1 \\times S_5.\u00a0 We have:
        \nI(t) = 1+3 t+\\frac{21 t^2}{2}+\\frac{55 t^3}{2}+\\frac{495 t^4}{8}+\\frac{4761 t^5}{40}+\\frac{48073 t^6}{240}+\\frac{502741 t^7}{1680} + \\cdots
        \nThe regularized period is:
        \n1+12 t^2+30 t^3+396 t^4+2160 t^5+20370 t^6+149520 t^7 + \\cdots
        \nThis is period sequence 64.<\/li>\n
      2. \\PP^1 \\times S_4.\u00a0 We have:
        \nI(t) = 1+4 t+19 t^2+\\frac{212 t^3}{3}+\\frac{2669 t^4}{12}+\\frac{8953 t^5}{15}+\\frac{251009 t^6}{180}+\\frac{908147 t^7}{315} + \\cdots
        \nThe regularized period is:
        \n1+22 t^2+96 t^3+1434 t^4+12480 t^5+148900 t^6+1606080 t^7 + \\cdots
        \nThis is period sequence 71.<\/li>\n
      3. \\PP^1 \\times S_3.\u00a0 We have:
        \nI(t) = 1+6 t+46 t^2+286 t^3+1489 t^4+\\frac{32939 t^5}{5}+\\frac{4550189 t^6}{180}+\\frac{8983549 t^7}{105}+ \\cdots
        \nThe regularized period is:
        \n1+56 t^2+492 t^3+10536 t^4+168600 t^5+3180980 t^6+58753800 t^7+ \\cdots
        \nThis is period sequence 45.<\/li>\n
      4. [Not very Fano] \\PP^1 \\times S_2.\u00a0 We have:
        \nI(t) = 1 + 12 t + 211 t^2 + 3092 t^3 + \\frac{150991}{4} t^4 + \\frac{1955353}{5} t^5 + \\frac{631426241}{180} t^6 + \\frac{2909156483}{105} t^7 + \\cdots
        \nThe regularized period is:
        \n1 + 278 t^2 + 6816 t^3 + 317850 t^4 + 12989760 t^5 + 578870180 t^6 + 26074520640 t^7 + \\cdots
        \nThis period sequence is non-Gorenstein.<\/li>\n
      5. [Not very Fano] \\PP^1 \\times S_1.\u00a0 We have:
        \nI(t) = 1 + 60 t + 6931 t^2 + 680740 t^3 + \\frac{223120591}{4} t^4 + 3882496633 t^5 + \\frac{8425548483661}{36} t^6 + \\frac{260867461874483}{21} t^7\\cdots
        \nThe regularized period is:
        \n1 + 10262 t^2 + 2021280 t^3 + 618997146 t^4 + 184490852160 t^5 + 57894898611620 t^6 + 18577980262739520 t^7 + \\cdots
        \nThis period sequence is non-Gorenstein.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        Rank 4 Fanos This is divisor of degree (1,1,1,1) on Regularizing gives period sequence 3: the blow-up of the cone over a smooth quadric surface in with center the disjoint union of the vertex and an elliptic curve on .\u00a0 The blow-up of the cone over with center the vertex is the toric variety with […]<\/p>\n","protected":false},"author":4,"featured_media":0,"parent":277,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-268","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/268","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\/4"}],"replies":[{"embeddable":true,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=268"}],"version-history":[{"count":49,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/268\/revisions"}],"predecessor-version":[{"id":5662,"href":"http:\/\/coates.ma.ic.ac.uk\/fanosearch\/index.php?rest_route=\/wp\/v2\/pages\/268\/revisions\/5662"}],"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=268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}