Higher rank Fano 3-folds

Rank 4 Fanos

  1. This is divisor of degree (1,1,1,1) on \PP^1 \times \PP^1 \times \PP^1 \times \PP^1
    I_X = \sum_{a,b,c,d \geq 0} \frac{(a+b+c+d)!}{a!^2b!^2c!^2d!^2} t^{a+b+c+d}
    Regularizing gives period sequence 3:
    1+12 t^2+48 t^3+540 t^4+4320 t^5 + \cdots
  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.  The blow-up of the cone over S with center the vertex is the toric variety with weight data:
    \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}
    The 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.  Thus X is the hypersurface u x + v f_{2,2}(s,t) = 0 in the toric variety with weight data:
    \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}
    Quantum Lefschetz gives:
    I_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}
    and regularizing gives period sequence 32:
    1+10 t^2+24 t^3+318 t^4+1680 t^5 + \cdots
  3. The blow-up of \PP^1 \times \PP^1 \times \PP^1 with center a curve \Gamma of tridegree (1,1,2).  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}.  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.  Now, in \PP^3 \times \PP^1, \Gamma becomes the complete intersection defined by equations:
    \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}
    Note that the second equation here is just the equation of X inside \PP^2 \times \PP^1.  Thus we can blow up \PP^3 \times \PP^1 along the locus:
    \begin{cases} u_2 - u_3 = 0 \\ u_0 z_1 - u_1 z_0 = 0 \end{cases}
    and then construct X by imposing the strict transform of the remaining equation u_0 u_1 - u_2 u_3 = 0.  This exhibits X as a complete intersection of type (L+M+N) \cdot (2L) inside the toric variety with weight data:
    \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}
    Note 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.  Quantum Lefschetz (which does not fail) gives:
    I_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}
    and regularizing gives period sequence 122:
    1+8 t^2+24 t^3+216 t^4+1320 t^5+ \cdotsA cleaner and more systematic development is as follows.  The curve \Gamma is defined scheme-theoretically by the equations:
    \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}
    inside \PP^1_{x_0,x_1} \times \PP^1_{y_0,y_1} \times \PP^1_{z_0,z_1}.  (Please look at the parametrization given above.)  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:
    \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}
    Now Quantum Lefschetz gives:
    I_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}
    and regularizing gives period sequence 122:
    1+8 t^2+24 t^3+216 t^4+1320 t^5+ \cdots
  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.  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.  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:
    \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}
    The 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.  (Note that X is rank 4 even though the ambient space is rank 3; there is no contradiction here because 2L is not ample.)  Quantum Lefschetz gives:
    I_X = \sum_{l, m, n \geq 0} \frac{(2l)!}{l!l!m!n!n!(l-m)!(m-n)!} t^{l+m+n}
    and regularizing gives period sequence 103:
    1+6 t^2+24 t^3+138 t^4+960 t^5+ \cdots
  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).  This is a hypersurface of type 2A+C+D in the toric variety with weight data:
    \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}
    Quantum Lefschetz gives:
    I_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}
    and regularizing gives period sequence 147:
    1+8 t^2+18 t^3+192 t^4+960 t^5+ \cdots
  6. the blow-up of \PP^1 \times \PP^1 \times \PP^1 with center the curve of tridegree (1,1,1).  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
    \rk \begin{pmatrix} x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \end{pmatrix} < 2
    Thus the blow-up X is cut out of the toric  variety with weight data:
    \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}
    by the equation:
    \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
    Quantum Lefschetz gives:
    I_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}
    and regularizing gives period sequence 65:
    1+6 t^2+18 t^3+114 t^4+720 t^5 + \cdots
  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).  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}.  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.  Thus X is a hypersurface of type A+B in the toric variety with weight data:
    \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}
    Quantum Lefschetz gives:
    I_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}
    and regularizing gives period sequence 69:
    1+6 t^2+12 t^3+114 t^4+480 t^5 + \cdots
  8. the blow-up of \PP^1 \times \PP^1 \times \PP^1 with center a curve \Gamma of tridegree (0,1,1).  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
    \begin{cases} z_0 = 0 \\ x_0 y_0 + x_1 y_1 = 0 \end{cases}
    and so X is cut out of the toric variety with weight data:
    \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}
    Quantum Lefschetz gives:
    I_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}
    and regularizing gives period sequence 105:
    1+6 t^2+12 t^3+90 t^4+480 t^5 + \cdots
  9. This is the toric variety with weight data:
    \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}
    Its regularized period sequence is period sequence 102:
    1+4 t^2+12 t^3+60 t^4+300 t^5 + \cdots
  10. This is the toric variety with weight data:
    \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}
    Its regularized period sequence is period sequence 142:
    1+6 t^2+6 t^3+90 t^4+240 t^5 + \cdots
  11. This is the toric variety with weight data:
    \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}
    Its regularized period sequence is period sequence 93:
    1+4 t^2+12 t^3+36 t^4+300 t^5 + \cdots
  12. This is the toric variety with weight data:
    \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}
    Its regularized period sequence is period sequence 150:
    1+4 t^2+6 t^3+60 t^4+120 t^5 + \cdots
  13. (degree 26, see the erratum)
    This is the blow-up of \PP^1 \times \PP^1 \times \PP^1 along a curve \Gamma of tridegree (1,1,3).  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:
    \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}
    We have -K_X = B+C+D.  This is ample on X but only semi-positive on the ambient space F.  Thus we are in the situation described here, and we use the same notation.  We have:
    \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}
    Inverting the mirror map gives:
    \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}
    Thus the cohomological-degree-zero part of the J-function of X is:
    \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}
    We 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:
    \exp(-3t) \sum_{a,b,c \geq 0} t^{a+b+2c} {(2a+b+c)! \over a!a!b!b!c!c!(a+c)!}
    and then doing the trick with factorials:
    I_{reg}(t) = 1+12 t^2+42 t^3+468 t^4+3360 t^5+31350 t^6 + \cdots
    This is period sequence 88.

Rank 5 Fanos

  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.  To compute this, take the defining equation of the quadric to be
    x_0 x_1 + x_1 x_2 + x_2 x_0 + x_3 x_4
    in \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\} .  Note that the conic in $\Pi$ contains the co-ordinate points [1:0:0], [0:1:0], [0:0:1].  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:
    \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}
    The regularized period sequence is:
    1+10 t^2+42 t^3+342 t^4+2640 t^5+21250 t^6+180600 t^7 + \cdots
    This is period sequence 114.
  2. This is the toric variety with weight data:
    \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}
    The regularized period sequence is:
    1+6 t^2+18 t^3+114 t^4+660 t^5+3930 t^6+25620 t^7+ \cdots
    This is period sequence 87.
  3. \PP^1 \times S_6
    The I-function is the product:
    I_{\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
    and the regularized period sequence is:
    1+8 t^2+12 t^3+168 t^4+600 t^5+5300 t^6+27720 t^7+ \cdots
    This is period sequence 43.

Rank > 5 Fanos

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

  1. \PP^1 \times S_5.  We have:
    I(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
    The regularized period is:
    1+12 t^2+30 t^3+396 t^4+2160 t^5+20370 t^6+149520 t^7 + \cdots
    This is period sequence 64.
  2. \PP^1 \times S_4.  We have:
    I(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
    The regularized period is:
    1+22 t^2+96 t^3+1434 t^4+12480 t^5+148900 t^6+1606080 t^7 + \cdots
    This is period sequence 71.
  3. \PP^1 \times S_3.  We have:
    I(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
    The regularized period is:
    1+56 t^2+492 t^3+10536 t^4+168600 t^5+3180980 t^6+58753800 t^7+ \cdots
    This is period sequence 45.
  4. [Not very Fano] \PP^1 \times S_2.  We have:
    I(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
    The regularized period is:
    1 + 278 t^2 + 6816 t^3 + 317850 t^4 + 12989760 t^5 + 578870180 t^6 + 26074520640 t^7 + \cdots
    This period sequence is non-Gorenstein.
  5. [Not very Fano] \PP^1 \times S_1.  We have:
    I(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
    The regularized period is:
    1 + 10262 t^2 + 2021280 t^3 + 618997146 t^4 + 184490852160 t^5 + 57894898611620 t^6 + 18577980262739520 t^7 + \cdots
    This period sequence is non-Gorenstein.

3 Comments

  1. Sergey says:

    How to make history of this and two other pages trackable?

  2. Tom says:

    I’m not sure. You can use the RSS feeds for posts and comments, but I don’t know how to check easily (i.e. via RSS or similar) that a page has been edited. Do you use Twitter? If you like I can set things up so that you get a tweet every time I make a significant change to a page.

  3. Sergey says:

    Ok, I just registered there with username ‘serugey’

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