Higher rank Fano 3-folds

Rank 4 Fanos

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:
$1+12 t^2+48 t^3+540 t^4+4320 t^5 + \cdots$
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:
$1+10 t^2+24 t^3+318 t^4+1680 t^5 + \cdots$
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:
$1+8 t^2+24 t^3+216 t^4+1320 t^5+ \cdots$ A 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:
$1+8 t^2+24 t^3+216 t^4+1320 t^5+ \cdots$
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:
$1+6 t^2+24 t^3+138 t^4+960 t^5+ \cdots$
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:
$1+8 t^2+18 t^3+192 t^4+960 t^5+ \cdots$
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:
$1+6 t^2+18 t^3+114 t^4+720 t^5 + \cdots$
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:
$1+6 t^2+12 t^3+114 t^4+480 t^5 + \cdots$
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:
$1+6 t^2+12 t^3+90 t^4+480 t^5 + \cdots$
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:
$1+4 t^2+12 t^3+60 t^4+300 t^5 + \cdots$
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:
$1+6 t^2+6 t^3+90 t^4+240 t^5 + \cdots$
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:
$1+4 t^2+12 t^3+36 t^4+300 t^5 + \cdots$
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:
$1+4 t^2+6 t^3+60 t^4+120 t^5 + \cdots$
(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

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.
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.
$\PP^1 \times S_6$
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.

$\PP^1 \times S_5$ . 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.
$\PP^1 \times S_4$ . 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.
$\PP^1 \times S_3$ . 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.
[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.
[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.

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3 Comments

Sergey says:

September 7, 2010 at 12:40 pm

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

September 9, 2010 at 3:02 pm

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.
Sergey says:

September 9, 2010 at 6:26 pm

Ok, I just registered there with username ‘serugey’

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