Fano Varieties and Extremal Laurent Polynomials

Rank 3 Fano 3-folds

a double cover of $\PP^1 \times \PP^1 \times \PP^1$ branched along a divisor of tridegree (2,2,2). This is a hypersurface of type $2L+2M+2N$ in the toric variety with weight data:
$\begin{array}{cccccccc} s_0 & s_1 & t_0 & t_1 & u_0 & u_1 & y & \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & L \\ 0& 0 & 1 & 1 & 0 & 0 & 1 & M\\ 0 & 0 & 0 & 0& 0 &0 & 1 & N \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(2l+2m+2n)! \over l! l!m! m!n!n!(l+m+n)!}$
and regularizing gives period sequence 22:
$1+54 t^2+672 t^3+15642 t^4+336960 t^5 + \cdots$
Note that this is a G-Fano.
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 $|L|$ . As discussed here, this is period sequence 97.
We do not understand this variety: 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:
$\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & s_0 & s_1 & t & \\ 1 & 1 & 0 & 0 & -1 & -1 & 0 & A \\ 0& 0 & 1 & 1 & -1 & -1 & 0 & B\\ 0 & 0 & 0 & 0& 1 &1 & 1 & C \end{array}$
The line bundle $L$ is ${-A}-B+C$ , so $X$ is a member of $|B+2C|$ and $-K_X = C-B$ . The equation defining $X$ takes the form:
$t^2 a_1(y) + s_0 t b_{1,2}(x,y) + s_1 t c_{1,2}(x,y) + s_0^2 d_{2,3}(x,y) + s_0 s_1 e_{2,3}(x,y) + s_1^2 f_{1,2}(x,y) = 0$
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 $y_j$ Consider now the subvariety $S$ defined by the equations $s_0 = s_1 = 0$ in $F$ . This is a copy of $\PP^1_{x_0,x_1} \times \PP^1_{y_0,y_1}$ ; note that $t=1$ on $S$ . The variety $X$ meets $S$ in the curve $\Gamma$ cut out by the equation $a_1(y) = 0$ inside $S$ . Without loss of generality we can take $a_1(y) = y_0$ , so that on $\Gamma$ we have $y_0 = 0$ and $y_1 = 1$ . The curve $\Gamma$ is a copy of $\PP^1$ . 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 $\Gamma$ ). Thus $-K_X = C-B$ is trivial on $\Gamma$ , and so $X$ is not Fano.
a divisor on $\PP^1 \times \PP^1 \times \PP^2$ of tridegree $(1,1,2)$ . This is straightforward quantum Lefschetz:
$I_X(t) = \sum_{k,l,m \geq 0} t^{k+l+m} {(k+l+2m)! \over (k!)^2 (l!)^2 (m!)^3}$
Regularizing this gives period sequence 31:
$1+20 t^2+132 t^3+1812 t^4+21720 t^5 + \cdots$
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 $Y \to \PP^1 \times \PP^2 \to \PP^2$ . The variety $Y$ is a hypersurface of type $2L+2M$ in the rank-2 toric variety with weight data:
$\begin{array}{ccccccc} s_0 & s_1 & x_0 & x_1 & x_2 & y & \\ 1 & 1 & 0 & 0 & 0 & 1 & L \\ 0& 0 & 1 & 1 & 1 & 1 & M \end{array}$
We 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:
$\begin{array}{cccccccc} s_0 & s_1 & x_0 & t_1 & t_2 & y & x & \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & L \\ 0& 0 & 1 & 0 & 0 & 1 & 1& M \\ 0 & 0 & 0 & 1 & 1 & 0 & -1 & N \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(2l+2m)! \over l! l! m! n!n!(l+m)!(m-n)!}$
and regularizing gives period sequence 151:
$1+24 t^2+156 t^3+2280 t^4+27960 t^5 + \cdots$
the blow-up of $\PP^1 \times \PP^2$ with center a curve of bidegree (5,2) that projects isomorphically to a conic in $\PP^2$ . We construct this as a codimension-2 complete intersection in $F = \PP(\cO\oplus\cO\oplus\cO(-1,-1) \to \PP^1 \times \PP^2$ . $F$ has weight data:
$\begin{array}{ccccccccc} t_0 & t_1 & y_0 & y_1 & y_2 & x_0 & x_1 & x & \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & L \\ 0& 0 & 1 & 1 & 1 & 0 & 0 & -1 & M\\ 0 & 0 & 0 & 0& 0 &1 & 1 & 1 & N \end{array}$
The equations defining $X$ are
$\begin{pmatrix} y_0 & y_1 & t_0 A_2(y) \\ y_1 & y_2 & t_1 B_2(y) \end{pmatrix} \cdot \begin{pmatrix} x_0 \\ x_1\\ x \end{pmatrix} = 0$
where $A_2, B_2$ are quadratic polynomials in the $y_i$ , and so $X$ is a complete intersection of type $(M+N)\cdot(M+N)$ in $F$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+n} {(m+n)! (m+n)! \over l! l!m! m!m!n!n!(n-m-l)!}$
and regularizing gives a period sequence that we do not have yet:
$1+32 t^2+204 t^3+3348 t^4+41040 t^5 + \cdots$
the blow-up of $\PP^3$ with center the disjoint union of a line $L$ and an elliptic curve $\Gamma$ of degree 4. Since $\Gamma$ is a (2,2) complete intersection in $\PP^2$ , our variety $X$ is a hypersurface of type $2L+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} s_0 & s_1 & x & x_2 & x_3 & t_0 & t_1 & \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 & L\\1 & 1 & -1 & 0 & 0 & 0 & 0 & M\\0 & 0 & 0 & 0 & 0 & 1 & 1 & N \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(2l+n)! \over m! m! (l-m)!l!l!n!n!}$
and regularizing gives period sequence 146:
$1+14 t^2+66 t^3+762 t^4+6960 t^5+ \cdots$
the blow-up $X$ of $W$ with center an elliptic curve which is a complete intersection of two members of $|-{1 \over 2} K_W|$ . 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 $(0,1,1)\cdot(1,1,1)$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(m+n)! (l+m+n)! \over l! l!m! m!m!n!n!n!}$
and regularizing gives period sequence 36:
$1+10 t^2+48 t^3+438 t^4+3720 t^5+ \cdots$

Note that this is a D4 form (i.e. the Picard–Fuchs equation has unexpectedly low degree in D). It is almost certainly a G-Fano, as there is an obvious $\ZZ/2\ZZ$ -action.
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. The weight data for $\mathbb{F}_1 \times \PP^2$ are:
$\begin{array}{cccccccc} x_0 & s_1 & s_2 & x & y_0 & y_1 & y_2 & \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & L\\0 & 1 & 1 &- 1 & 0 & 0 & 0 & M\\0 & 0 & 0 & 0 & 1 & 1 & 1 & N \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(l+2n)! \over l! m! m!(l-m)!n!n!n!}$
and regularizing gives period sequence 85:
$1+12 t^2+54 t^3+540 t^4+4620 t^5 + \cdots$
$\mathbb{F}_1 \times \PP^2$ admits an obvious map to $\PP^1_{s_1,s_2} \times \PP^2_{y_0,y_1,y_2}$ . 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). 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 $(0,2)\cdot(1,2)$ .
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 $\PP^2 \cong R_4$ . Thus $X$ is a hypersurface of type $4L+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} s_0 & s_1 & s_2 & x & y & s & t & \\ 1 & 1 & 1 & 0 & -2 & 4 & 0 & L\\0 & 0 & 0 & 1 & 1 & -1 & 0 & M\\0 & 0 & 0 & 0 & 0 & 1 & 1 & N \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(4l+n)! \over l! l! l!m!(-2l+m)!(4l-n+m)!n!}$
and regularizing gives period sequence 68:
$1+2 t^2+36 t^3+198 t^4+840 t^5+ \cdots$
the blow-up of a quadric $Q$ in $\PP^4$ with center a disjoint union of two conics on $Q$ . 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 $Q$ . So we can construct the variety $X$ as a hypersurface of type $2L$ in the toric variety with weight data:
$\begin{array}{cccccccc} s_0 & s_1 & t_2 & t_3 & x_4 & x & y & \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & L\\1 & 1 & 0 & 0 & 0 & -1 & 0 & M\\0 & 0 & 1 & 1 & 0 & 0 & -1 & N \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(2l)! \over m!m! n!n!l!(l-m)!(l-n)!}$
and regularizing gives period sequence 67:
$1+10 t^2+36 t^3+366 t^4+2640 t^5+ \cdots$
the blow-up $X$ of $V_7$ with center a complete intersection of two general members of ${-1/2} K_{V_7}$ . $X$ 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 $(1,1)\cdot(1,2)$ . Consider the toric variety with weight data:
$\begin{array}{cccccccc} t_0 & t_1 & x_0 & s_1 & s_2 & s_3 & x & \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & L\\0 & 0 & 0 & 1 & 1 & 1 & -1 & M\\1 & 1 & 0 & 0 & 0 & 0 & 0 & N \end{array}$
This is $\PP^1 \times V_7$ , and $X$ is cut out here as a hypersurface of type $L+M+N$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(l+m+n)! \over n! n! l!m!m!m!(l-m)!}$
and regularizing gives period sequence 107:
$1+6 t^2+30 t^3+186 t^4+1380 t^5+ \cdots$

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 $\PP^1_{t_0,t_1} \times \PP^2_{s_0,s_1,s_2}$ . 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). This suffices.
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:
$\rk \begin{pmatrix} x_0 & x_1 &x_ 2\\ x_1 & x_2 & x_3 \end{pmatrix} <2$
The 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:
$\begin{pmatrix} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \end{pmatrix} \cdot \begin{pmatrix} y_0 \\ y_1\\y_2 \end{pmatrix} = 0$

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:
$\begin{array}{ccccccccc} s_0 & x_1 & x_2 & s_3 & x & y_0 & y_1 & y_2 & \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & L \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & M\\ 1& 0 & 0 & 1 & -1 & 0 & 0 & 0 & N \end{array}$
The equations defining $X$ inside $F$ are:
$\begin{pmatrix} s_0 x & x_1 & x_2 \\ x_1 & x_2 & s_3 x \end{pmatrix} \cdot \begin{pmatrix} y_0 \\ y_1\\y_2 \end{pmatrix} = 0$
and so $X$ is a complete intersection of type $(L+M)\cdot(L+M)$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(l+m)!(l+m)! \over n! l!l!n!(l-n)!m!m!m!}$
and regularizing gives period sequence 144:
$1+8 t^2+30 t^3+240 t^4+1740 t^5+ \cdots$

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 $[x_0:x_1:x_2:s_3:x:y_0:y_1:y_2] \mapsto [s_0:s_3:y_0:y_1:y_2]$ . Rewriting the equations of $X$ as:

$\begin{pmatrix} s_0 y_0 & y_1 & y_2 \\ s_3 y_2 & y_0 & y_1 \end{pmatrix} \cdot \begin{pmatrix} x \\ x_1\\x_2 \end{pmatrix} = 0$
we see that $X$ is the blow-up of $\PP^1 \times \PP^2$ along the curve $C$ given by:
$\rk \begin{pmatrix} s_0 y_0 & y_1 & y_2 \\ s_3 y_2 & y_0 & y_1 \end{pmatrix} < 2$
The equations defining $C$ are:
$\begin{cases} y_1^2 - y_0 y_2 = 0 \\ s_0 y_0^2 - s_3 y_1 y_2 = 0 \\ s_0 y_0 y_1 - s_3 y_2^2 = 0\end{cases}$
and so $C$ lies entirely within the “cylinder surface” $y_1^2 - y_0 y_2 = 0$ . This cylinder surface is abstractly isomorphic to $\PP^1_{s_0,s_3} \times \PP^1_{t_0,t_1}$ , where $y_0 = t_0^2, y_1 = t_0 t_1, y_2 = t_1^2$ . The equations of $C$ become:
$\begin{cases} t_0(s_0 t_0^3 - s_3 t_1^3) = 0 \\ t_1(s_0 t_0^3 - s_3 t_1^3) = 0\end{cases}$
Thus $C$ is as described above.
the blow-up…
the blow-up of $\PP^3$ with center the union of a cubic in a plane $\Pi$ and a point $P$ not in $\Pi$ . Let $\Pi$ be $x_0 = 0$ in $\PP^3_{x_0,x_1,x_2,x_3}$ , and let $P$ be $x_1 = x_2 = x_3 = 0$ . Thus $X$ is the blow-up of the curve $\Gamma \subset V_7$ given by:
$\begin{cases} x_0 = 0 \\ a_3(s_1,s_2,s_3) = 0 \end{cases}$
where the blow-up $V_7 \to \PP^3$ is $[x_0:s_1:s_2:s_3:x] \mapsto [x_0:s_1 x: s_2 x: s_3 x]$ . Thus $X$ is a hypersurface of type $3M+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} x_0 & s_1 & s_2 & s_3 & x & s & t & \\ 1 & 0 & 0 & 0 & 1 & -1 & 0 & L \\ 0 & 1 & 1 & 1 & -1 & 3 & 0 & M\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+2m+n} {(3m+n)! \over l!m!m!m!(l-m)!(3m+n-l)!n!}$
and regularizing gives period sequence 148:
$1+2 t^2+18 t^3+102 t^4+420 t^5+ \cdots$
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 $Q$ . $X$ is the blow-up of $\PP^1 \times \PP^2$ along a curve of bidegree (2,2) that is a complete intersection of type $(0,2)\cdot(1,1)$ . To see this, we first exhibit $X$ as a hypersurface in a toric variety. 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:
$\begin{array}{cccccccc} s_0 & s_1 & s_2 & t_3 & t_4 & x & y & \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & L \\ 1 & 1 & 1 & 0 & 0 & -1 & 0 & M\\ 0 & 0 & 0 & 1 & 1 & 0 & -1 & N\end{array}$
The 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 $[t_0 : t_1 : s_0 : s_1 : s_2]$ . $X$ is cut out of the above toric variety by a section of $L+M$ . Thus we have:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(l+m)! \over m!m!m!n!n!(l-m)!(l-n)!}$
and regularizing gives period sequence 112:
$1+6 t^2+18 t^3+138 t^4+780 t^5+ \cdots$
To see that $X$ is the blow-up of $\PP^1 \times \PP^2$ as claimed, write the equation defining $X$ as $x a + y b = 0$ . Then by homogeneity $a$ is a section of (0,2) and $b$ is a section of (1,1).
the blow-up of $B_7$ with center the strict transform of a twisted cubic through the blown-up point $P \in \PP^3$ . 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
$\rk \begin{pmatrix} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \end{pmatrix} < 2$
Let the blow-up $B_7 \to \PP^3$ be given by $[x_0 : s_ 1 : s_2 : s_3 : x] \mapsto [x_0 : s_1 x: s_2 x: s_3 x]$ . Then the strict transform of $\Gamma$ in $B_7$ is given by:
$\rk \begin{pmatrix} x_0 & s_1 & s_2 \\ x s_1 & s_2 & s_3 \end{pmatrix} < 2$
As before, we introduce new variables $y_0, y_1, y_2$ and the toric variety $F$ with weight data:
$\begin{array}{ccccccccc} x_0 & s_1 & s_2 & s_3 & x & y_0 & y_1 & y_2 & \\ 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & L \\ 0 & 1 & 1 & 1 & -1 & 0 & -1 & -1 & M \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & N \end{array}$
$X$ is cut out by the equations:
$\begin{cases} x_0 y_0 + s_1 y_1 + s_2 y_2 = 0 \\ s_1 x y_0 + s_2 y_1 + s_3 y_2 = 0\end{cases}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{2l+n} {(l+n)!(l+n)! \over l!m!m!m!(l-m)!n!(l-m+n)!(l-m+n)!}$
and regularizing gives period sequence 119:
$1 + 4 t^2 + 18 t^3 + 84 t^4 + 540 t^5 + \cdots$
a smooth divisor on $\PP^1 \times \PP^1 \times \PP^2$ of tridegree $(1,1,1)$ . This is straightforward quantum Lefschetz:
$I_X(t) = \sum_{k,l,m \geq 0} t^{k+l+2m} {(k+l+m)! \over (k!)^2 (l!)^2 (m!)^3}$
Regularizing this gives period sequence 37:
$1 + 4 t^2 + 12 t^3 + 84 t^4 + 360 t^5 + \cdots$
Note that $X$ is also the blow-up of $\PP^1 \times \PP^2$ along a curve of bidegree (1,2).
the blow-up of $\PP^3$ with center a disjoint union of a line $L$ and a conic. 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 $x_0 x_1 + x_2^2 = x_3 = 0$ . 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 $x_3 = s_0 s_1 x^2 + x_2^2 = 0$ . Thus $X$ is a hypersurface of type $2L+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} s_0 & s_1 & x & x_2 & x_3 & s & t & \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & L \\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & M \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N \end{array}$
cut out by the equation $s x_3 = t(x_2^2+s_0 s_1 x^2)$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{2l+m+n} {(2l+n)! \over m!m!(l-m)l!l!(l+n)!n!}$
and regularizing gives period sequence 160:
$1+4 t^2+18 t^3+60 t^4+480 t^5+ \cdots$
the blow-up of a quadric $Q$ with center two non-colinear points. 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 $\PP^4$ . This is a hypersurface of type $2L$ in the toric variety with weight data
$\begin{array}{ccccccc} x_0 & x_1 & s_2 & s_3 & s_4 & x & \\ 1 & 1 & 0 & 0 & 0 & 1 & L \\ 0 & 0 & 1 & 1 & 1 & -1 & M \end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m \geq 0} t^{l+2m} {(2l)! \over l!l!m!m!m!(l-m)!}$
and regularizing gives period sequence 57:
$1+2 t^2+12 t^3+54 t^4+240 t^5+ \cdots$
the blow-up of a 3-dimensional quadric $Q$ with center two disjoint lines on it. 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 $x_2=x_3=x_4=0$ . Thus $X$ is a hypersurface of type $M+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} s_0 & s_1 & s_2 & s_3 & s & x & y & \\ 0 & 0 & 0 & 0 & -1 & 1 & 1 & L \\ 1 & 1 & 0 & 0 & 1 & -1 & 0 & M \\ 0 & 0 & 1 & 1 & 1 & 0 & -1 & N\end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m \geq 0} t^{l+m+n} {(m+n)! \over m!m!n!n!(-l+m+n)!(l-m)!(l-n)!}$
and regularizing gives period sequence 63:
$1+4 t^2+12 t^3+60 t^4+360 t^5+ \cdots$
the blow-up of $\PP^1 \times \PP^2$ along a curve of bidegree (2,1). $X$ 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
$\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & y_2 & s & t& \\ 1 & 1 & 0 & 0 & 0 & 0 & -1 & L \\ 0 & 0 & 1 & 1 & 1 & 0 & -1 & M \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {(m+n)! \over l!l!m!m!m!n!(n-l-m)!}$
and regularizing gives period sequence 98:
$1+6 t^2+6 t^3+114 t^4+240 t^5+ \cdots$
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
$\begin{array}{cccccccc} x_0 & x_1 & y_0 & y_1 & y_2 & s & t& \\ 1 & 1 & 0 & 0 & 0 & -1 & 0 & L \\ 0 & 0 & 1 & 1 & 1 & 0 & -2 & M \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{l+m+n} {n! \over l!l!m!m!m!(n-l)!(n-2m)!}$
and regularizing gives period sequence 152:
$1+2 t^2+6 t^3+54 t^4+180 t^5+ \cdots$
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 $x_1 = x_2 = x_3 =0$ . Define the conic by:
$\begin{cases} x_3 = 0 \\ x_0 x_1 + x_2^2 = 0\end{cases}$
Taking the proper transform under the blow-up $B_7 \to \PP^3$ gives the equations:
$\begin{cases} s_3 = 0 \\ x_0 s_1 + x s_2^2 = 0\end{cases}$
where 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:
$s(s_3) + t(x_0 s_1 + x s_2^2) = 0$
This is a hypersurface of type $L+M+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} x_0 & s_1 & s_2 & s_3 & x & s & t & \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & L \\ 0 & 1 & 1 & 1 & -1 & 0 & 0 & M\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{2l+m+n} {(l+m+n)! \over l!m!m!m!(l-m)!(l+n)!n!}$
and regularizing gives period sequence 158:
$1+2 t^2+12 t^3+30 t^4+180 t^5+\cdots$
the fiber product $X = W \times_{\PP^2} \mathbb{F}_1$ where $W$ is a (1,1) hypersurface in $\PP^2 \times \PP^2$ . This is the blow up of $\PP^1 \times \PP^2$ along a curve of bidegree (1,1). 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:
$\begin{cases} y_0 x_0 + y_1 x_1 + y_2 x_2 = 0 \\ s_0 x_0 + s_1 x_1 = 0 \end{cases}$
The 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 $\PP^2$ .

We now exhibit $X$ as the blow-up of a curve in $\PP^1 \times \PP^2$ by projecting to $\PP^2_{y_0,y_1,y_2} \times \PP^1_{s_0,s_1}$ . This projection is an isomorphism away from the locus where the matrix
$\begin{pmatrix} y_0 & y_1 & y_2 \\ s_0 & s_1 & 0 \end{pmatrix}$
drops rank. This locus is:
$\begin{cases} y_2 = 0 \\ y_0 s_1 - y_1 s_0 = 0\end{cases}$
i.e. a curve in $\PP^1 \times \PP^2$ of bidegree (1,1).
We can further simplify things by writing $X$ as a hypersurface in $\PP^2 \times \mathbb{F}_1$ . 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:
$t_0 x y_0 + t_1 x y_1 + x_2 y_2 = 0$
Thus $X$ is a hypersurface of type $L+N$ in the toric variety with weight data:
$\begin{array}{cccccccc} y_0 & y_1 & y_2 & s_0 & s_1 & x & x_2 & \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & L \\ 0 & 0 & 0 & 1 & 1 & -1 & 0 & M \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & N\end{array}$
Quantum Lefschetz gives:
$I_X = \sum_{l,m,n \geq 0} t^{2l+m+n} {(l+n)! \over l!l!l!m!m!(n-m)!n!}$
and regularizing gives period sequence 86:
$1+4 t^2+6 t^3+60 t^4+180 t^5 + 1210 t^6 + 5460 t^7 + 30940 t^8 + 165480 t^9 +\cdots$
the blow-up of $\PP^3$ with center two disjoint lines. This is the toric variety with weight data:
$\begin{array}{ccccccc} s_0 & s_1 & t_0 & t_1 & x & y & \\ 0 & 0 & 0 & 0 & 1 & 1 & L \\ 1 & 1 & 0 & 0 & -1 & 0 & M\\ 0 & 0 & 1 & 1 & 0 & -1 & N\end{array}$
The blow-up map is $[s_0 : s_1 : t_0 : t_1 : x : y] \mapsto [s_0 x : s_1 x : t_0 y : t_1 y ]$ . We have:
$I_X = \sum_{l,m,n \geq 0} t^{2l+m+n} {1 \over m!m!n!n!(l-m)!(l-n)!}$
and regularizing gives period sequence 41:
$1+2 t^2+12 t^3+30 t^4+120 t^5 + \cdots$
the blow-up $X$ of $\PP^3$ with center a disjoint union of a point and a line. This is also the blow-up of $\PP^1 \times \PP^2$ in a curve of bidegree (0,1). So $X$ is a toric variety, with weight data:
$\begin{array}{ccccccc} s_0 & x_1 & t_0 & y_1 & y_2 & x & \\ 1 & 1 & 0 & 0 & 0 & 0 & L \\ 0 & 0 & 1 & 1 & 1 & 0 & M\\ -1 & 0 & -1 & 0 & 0 & 1 & N\end{array}$
The blow-up map is $[s_0 : x : t_0 : y_1 : y_2 : x] \mapsto [s_0 x : x_1 : t_0 x : y_1 : y_2]$ . We have:
$I_X = \sum_{l,m,n \geq 0} t^{2l+3m-n} {1 \over (l-n)!l!(m-n)!m!m!n!}$
and regularizing gives period sequence 113:
$1+2 t^2+6 t^3+30 t^4+120 t^5 + \cdots$
$X = \PP^1 \times \PP^1 \times \PP^1$ . This gives:
$I_X = \sum_{k,l,m \geq 0} t^{2k+2l+2m} {1 \over k!k!l!l!m!m!}$
and regularizing gives period sequence 21:
$1 + 6 t^2 + 90 t^4 + 0 t^5 + \cdots$
$X = \PP^1 \times \mathbb{F}_1$ . This gives:
$I_X = \sum_{k,l,m \geq 0} t^{2k+l+2m} {1 \over k!k!l!l!(m-l)!m!}$
and regularizing gives period sequence 90:
$1+4 t^2+6 t^3+36 t^4+180 t^5 + \cdots$
the blow-up of $B_7$ with center a line on the exceptional divisor of the blow-up $B_7 \to \PP^3$ . This is a toric variety with weight data:
$\begin{array}{ccccccc} x_0 & s_1 & s_2 & s & x & y & \\ 1 & 0 & 0 & 0 & 1 & 0 & L \\ 0 & 1 & 1 & 1 & -1 & 0 & M\\ 0 & 0 & 0 & 1 & 1 & -1 & N\end{array}$
We have:
$I_X = \sum_{l,m,n \in \ZZ} t^{2l+2m+n} {1 \over l!m!m!(m+n)!(l-m+n)!(-n)!}$
and regularizing gives period sequence 163:
$1+2 t^2+30 t^4+60 t^5+ \cdots$
the blow-up of $B_7$ with center the strict transform of a line through the blown-up point $P \in \PP^3$ . This is a toric variety with weight data:
$\begin{array}{ccccccc} x_0 & s_1 & t_2 & t_3 & x & y & \\ 1 & 0 & 0 & 0 & 1 & 0 & L \\ 0 & 1 & 0 & 0 & -1 & 1 & M\\ 0 & 0 & 1 & 1 & 0 & -1 & N\end{array}$
We have:
$I_X = \sum_{l,m,n \geq 0} t^{2l+m+n} {1 \over l!m!n!n!(l-m)!(m-n)!}$
and regularizing gives period sequence 84:
$1+2 t^2+6 t^3+30 t^4+60 t^5+ \cdots$
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:
$\begin{array}{ccccccc} 1 & 1 & 0 & 0 & 0 & 1 & L \\ 0 & 0 & 1 & 1 & 0 & 1 & M\\ 0 & 0 & 0 & 0 & 1 & 1 & N\end{array}$
We have:
$I_X = \sum_{l,m,n \geq 0} t^{3l+3m+2n} {1 \over l!l!m!m!n!(l+m+n)!}$
and regularizing gives period sequence 53:
$1 + 2 t^2 + 12 t^3 + 6 t^4 + 120 t^5+ \cdots$

Note

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

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5 Comments

Sergey says:

September 7, 2010 at 12:02 pm

On 25, 29, 30 and 31: these are smooth toric varieties.
They correspond to period sequences 41, 53, 84 and 163.

To be more precise:

25 is unique smooth toric Fano with P=3 and degree 44,
it corresponds to period sequence 41.

31 is unique smooth toric Fano with P=3 and degree 52,
it corresponds to period sequence 53.

29 and 30 has same degree 50 and correspond to p.s. 84 (grdb – 520136) and 163 (grdb – 520127).
Need an extra computation to separate these two.
Here is it.

Mirror of $\PP^3$ is $w_{\PP^3} = x+y+z+\frac{1}{xyz}$

$V_7$ is a blowup of point, its mirror is
$w_{V_7} = w_{\PP^3} + xyz = (x + y + z + \frac{1}{xyz}) +xyz$ .

2.29 is blowup of line on exceptional divisor, so its mirror is
$w_{2.29} = w_{V_7} + x^2 yz = ((x+y+z+\frac{1}{xyz}) + xyz) +x^2 yz$
This Laurent polynomial has period sequence 163
[1, 0, 2, 0, 30, 60, 380, 840, 5950, 22680]

2.30 is blowup of line that strict transform of line passing the center of blowup $V_7 \to \PP^3$ ,
so its mirror is
$w_{2.30} = w_{V_7} + xy = ((x+y+z+\frac{1}{xyz}) + xyz) + xy$
This Laurent polynomial has period sequence 84
[1, 0, 2, 6, 30, 60, 470, 1680, 7630, 34440]

[Sorry, I had a typo here (!), 84 is correct]

Also, for 2.25 mirror is $w_{2.25} = (x+y+z+\frac{1}{xyz}) + xy + \frac{1}{xy}$
Sergey says:

September 12, 2010 at 1:17 pm

On 3.7

Note that this is a D4 form (i.e. the Picard–Fuchs equation has unexpectedly low degree in D). It is almost certainly a G-Fano, as there is an obvious -action.

It is not a G-Fano, but has a little of symmetry ( $A_1$ ), so it looks like
Fano with Picard number 2.
(According to Matsuki’s data) other Fanos with this property in this list should have the following numbers:
3, 9, 10, 17, 19, 20, 25, 31

I wrote a wider review of these issues in the post
expected distribution of equations

—-

Also I propose not to call this type of equation D4, since D4 is already reserved for equations that look like RQDE of P^4 (or 4-dimensional quadric). Better name for RQDE
of general Fano 3-fold with Picard number 2 is D3+1.

In general, type of RQDE for generic Fano variety is classified by its Lefschetz decomposition i.e. partition or Young tableux.

For Fano threefolds we will probably have just D3, D3+1, D3+2 and four D3+3’s.
(3+2 is a nickname for 3+2×1,
3+3 is a nickname for 3+3×1).
Sergey says:

September 14, 2010 at 12:06 am

On 6, 10 and 23.

3.6 has period sequence 146
It is represented by complete intersection of degrees (1,0,2) and (0,1,1) in $\PP^1 \times \PP^1 \times \PP^3$ .

3.10 has period sequence 67.
It is represented by complete intersection of degrees (1,0,1), (0,1,1) and (0,0,2) in $\PP^1 \times \PP^1 \times \PP^4$ .

3.23 has period sequence 86.
It is represented by complete intersection of degrees (1,1,0) and (0,1,1) in $\PP^1 \times \PP^2 \times \PP^2$ .
Sergey says:

September 16, 2010 at 11:54 am

On 3.2: its description basically says it is a divisor in smooth toric fourfold.
Sergey says:

October 5, 2010 at 3:00 pm

On 3.24:

there is a typo in the very end (!)

the period is _not_ ps[130]: [1, 0, 4, 6, 60, 180, 1210, 5040, 30940, 150360]

but ps[86]: [1, 0, 4, 6, 60, 180, 1210, 5460, 30940, 165480]

They have same first 7 entries, so it was easy to mistake.

Moreover, period sequence 130 is bad – it has (2D-1) as a multiple. So this resolves the ambiguity we had before.

Also – 3.24 has 3 terminal Gorenstein degenerations,
and TGTD ansatz applied to them is exactly ps[86].

3.24 corresponds to ps[86]
3.23 should correspond to ps[158]

And there will be exactly 1-1 correspondence between 98 Fano threefolds and 98 good period sequences.

Following is the computation of first 10 terms:

> sum(l=0,10,sum(m=0,10,sum(n=m,10, t^(2*l+m+n) * (l+n)! / l!^3 /m!^2 /n! /(n-m)!))) +O(t^11)

1 + t + 5/2*t^2 + 19/6*t^3 + 109/24*t^4 + 581/120*t^5 + 3371/720*t^6 + 4021/1008*t^7 + 123229/40320*t^8 + 773029/362880*t^9 + 983333/725760*t^10 + O(t^11)

> reg(%)

1 + t + 5*t^2 + 19*t^3 + 109*t^4 + 581*t^5 + 3371*t^6 + 20105*t^7 + 123229*t^8 + 773029*t^9 + O(t^10)

> nor(%)

1 + 4*t^2 + 6*t^3 + 60*t^4 + 180*t^5 + 1210*t^6 + 5460*t^7 + 30940*t^8 + 165480*t^9 + O(t^10)

> Vec(%)
[1, 0, 4, 6, 60, 180, 1210, 5460, 30940, 165480]

Moreover, the description for 3.24 seems too complicated.
After on the third line X is described as the complete intersection of degrees (1,1,0) and (0,1,1) in $\PP^1 \times \PP^2 \times \PP^2$ we can already apply quantum Lefschetz:

$\sum_{a,b,c \geq 0} t^{a+b+2c} \frac{(a+b)! (b+c)!}{a!^2 b!^3 c!^3} = 1 + 2 t + 4 t^2 + 19/3 t^3 + 55/6 t^4 + 343/30 t^5 + 4477/360 t^6 + 3781/315 t^7 + 104959/10080 t^8 + 18611/2268 t^9 + \dots$ ,
regularizing and normalizing we get
$1 + 4 t^2 + 6 t^3 + 60 t^4 + 180 t^5 + 1210 t^6 + 5460 t^7 + 30940 t^8 + 165480 t^9 + \dots$
i.e. period sequence 86.