Fano Varieties and Extremal Laurent Polynomials » Rank 2 Fano 3-folds

Rank 2 Fano 3-folds

We compute the quantum period sequences of Fano 3-folds in the Mori-Mukai list.

[Not very Fano]The blow-up of $V_1$ with centre an elliptic curve which is the intersection of two members of $|{-{1/2}} K|$ . This is a hypersurface in a toric variety $F$ . The divisor diagram for $F$ is $\begin{array}{l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 2 & 3 \end{array}$ .
Note that $F$ is a scroll over $\CC P^1$ with fibre $\CC P(1,1,2,3)$ . There is a morphism $F \to \CC P(1,1,1,2,3)$ , which is the blow up along $x_0=x_1=0$ ; this map sends $[s_0,s_1,x, x_2, y, z]$ to $[s_0x,s_1x,x_2,y,z]$ . There are two line bundles $L, M$ on $F$ : $s_0,s_1$ are section of $L$ ; $xs_0, xs_1,x_2$ are sections of $M$ ; $y$ is a section of $2M$ and $z$ of $3M$ . $X$ is cut out by a section of $6M$ : we have $-K_X=L+M$ .

Applying quantum Lefschetz gives:
$I_X= \sum_{l, m\geq 0} t^{l+m}\frac{(6m)!}{l!l!m!(2m)!(3m)!\Gamma(1+m-l)}.$
Regularizing this (that is, pre-multiplying by $e^{-61t}$ so as to kill the linear term in $t$ , and then replacing $\sum a_kt^k$ by $\sum k! a_k t^k$ ) gives a period sequence that is not in our list:
$1+10380 t^2+2082840 t^3+650599740 t^4+199351017360 t^5\cdots$

Note that there are two birational models of the ambient space here, corresponding to the two chambers in the divisor diagram. But the sum defining the I-function really takes place over the intersection of the Mori cones of the two birational models, because of the factor of $1/\Gamma(1+m-l)$ in the summand. This factor vanishes outside one of the Kahler cones; similarly $1/l! = 1/\Gamma(1+l)$ vanishes outside the other Kahler cone. Similar things happen in many of the examples below.
[Not very Fano]The double cover of $\CC P^1 \times \CC P^2$ branched along a divisor of bidegree $(2,4)$ . This is a hypersurface in a toric variety $F$ with divisor diagram $\begin{array}{l l l l l l} 1 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 1 & 1 & 2 \end{array}$ . With coordinates $x_0, x_1,y_0,y_1,y_2 ,z$ , the defining equation of $X$ is $z^2=f_{2,4}(x_0,x_1;y_0,y_1,y_2)$ . Denote by $L$ the line bundle with sections $x_i$ and by $M$ the line bundle with sections $y_j$ ; then $-K_X = L+M$ .Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(2l+4m)!}{l! l! m! m! m! (l+2m)!}.$
Regularizing this gives a period sequence that is not in our list:
$1+470 t^2+21216 t^3+1562778 t^4+114717120 t^5+\cdots$
[Not very Fano]the blow up of $V_2$ with centre an elliptic curve which is the intersection of two elements of $|-1/2K|$ . This is a hypersurface in a toric variety $F$ . The divisor diagram for $F$ is $\begin{array}{l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 2 \end{array}$ .
Note that $F$ is a scroll over $\CC P^1$ with fibre $\CC P(1,1,1,2)$ . There is a morphism $F \to \CC P(1,1,1,1,2)$ , which is the blow up along $x_0=x_1=0$ ; this map sends $[s_0,s_1,x, x_2, x_3, y]$ to $[s_0x,s_1x,x_2,x_3,y]$ . There are two line bundles $L, M$ on $F$ : $s_0,s_1$ are section of $L$ ; $xs_0, xs_1,x_2, x_3$ are sections of $M$ ; $y$ is a section of $2M$ . $X$ is cut out by a section of $4M$ : we have $-K_X=L+M$ . Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(4m)!}{l! l! m! m! (2m)! \Gamma (1+m-l)}.$
Regularizing this gives a period sequence that is not in our list:
$1+300 t^2+8472 t^3+438588 t^4+21183120 t^5+\cdots$
the blow up of $P^3$ with centre an intersection of two cubics. Thus, $X$ is a divisor of bidegree $(1,3)$ on $\CC P^1 \times \CC P^3$ . We denote by $L$ and $M$ the pull backs of the tautological bundles on the two factors. We have $-K_X = L + M$ , and quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+3m)!}{l! l! m! m! m! m!}.$
Regularizing this gives period sequence 49:
$1+90 t^2+1518 t^3+46086 t^4+1327320 t^5+\cdots$
the blow up of $V_3$ with centre a plane cubic.This is a hypersurface in a toric variety $F$ . The divisor diagram for $F$ is $\begin{array}{l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 \end{array}$ .
Note that $F$ is a scroll over $\CC P^1$ with fibre $\CC P^3$ . There is a morphism $F \to \CC P^4$ , which is the blow up along $x_0=x_1=0$ ; this map sends $[s_0,s_1,x, x_2, x_3, x_4]$ to $[s_0x,s_1x,x_2,x_3,x_4]$ . There are two line bundles $L, M$ on $F$ : $s_0,s_1$ are section of $L$ ; $xs_0, xs_1,x_2, x_3,x_4$ are sections of $M$ . $X$ is cut out by a section of $3M$ : we have $-K_X=L+M$ .Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(3m)!}{l! l! m! m! m! \Gamma (1+m-l)}.$
Regularizing this gives period sequence 34:
$1 + 66t^2+816t^3+20214t^4+449640t^5+\cdots$ .
a divisor of bidegree $(2,2)$ in $\CC P^2 \times \CC P^2$ . Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(2l+2m)!}{l! l! l! m! m! m!}.$
Regularizing this gives period sequence 11:
$1+44 t^2+528 t^3+11292 t^4+228000 t^5+\cdots$ .
Note that this is a D3 form: even though the Fano $X$ has rank 2, the quantum cohomology D-module splits off a “rank 1” irreducible piece (i.e. a piece of dimension 4, which is the size of the cohomology of a rank-1 Fano 3-fold).
the blow up of a quadric $Q$ with centre the intersection of two members of $\cO (2)$ . Thus, $X$ is the complete intersection of two divisors in $\PP^1 \times \PP^4$ , of bidegrees $(0,2)$ and $(1,2)$ . We denote by $L$ and $M$ the pull backs of the tautological bundles on the two factors. We have $-K_X = L + M$ , and quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(2m)!(l+2m)!}{l! l! m! m! m! m! m!}.$
Regularizing this gives period sequence 51:
$1+36 t^2+348 t^3+6516 t^4+110880 t^5+\cdots$
a double cover of $V_7$ with branch locus a member $B$ of $|-K_{V_7}|$ such that $B\cap D$ is smooth, where $D$ is the exceptional divisor of the blow-up $V_7 \to \PP^3$ . This is a hypersurface in a toric variety $F$ . The divisor diagram for $F$ is $\begin{array}{llllll} 1 & 1 & 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{array}$ . Call the co-ordinates $s_0, s_1, s_2, x, x_3, z$ . Let $L$ be the line bundle with sections $s_0,s_1,s_2$ and let $M$ be the line bundle with sections $s_0 x, s_1 x, s_2 x, x_3$ ; $z$ is a section of $L+M$ . The variety $X$ is a section of $2L+2M$ on $F$ ; we have $-K_X = L+M$ .Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(2l+2m)!}{l! l! l! \Gamma(1+m-l) m! (l+m)!}.$
Regularizing this gives period sequence 26:
$1+26 t^2+216 t^3+3582 t^4+54480 t^5+\cdots$
the blow up of $\PP^3$ in a curve $\Gamma$ of degree 7 and genus 5. $\Gamma$ is cut by the equations:
$\rk \left( \begin{array}{lll} l_0 & l_1 & l_2 \\ q_0 & q_1 & q_2 \end{array} \right) < 2$
where the $l_i$ are linear forms and the $q_j$ are quadratics. Let $y_0= l_0 q_1 - l_1 q_0, y_1 = l_2 q_0-l_0 q_2, y_2 = l_0 q_1 - l_1 q_0$ . The relations (szyzgies) between these equations are generated by:
$\begin{cases} l_0 y_0 + l_1 y_1 + l_2 y_2 = 0 \\ q_0 y_0 + q_1 y_1 + q_2 y_2 = 0 \end{cases}$
Thus $X$ is given by these two equations in $\PP^3 \times \PP^2$ , where the first factor has co-ordinates $x_0, x_1, x_2, x_3$ and the second factor has co-ordinates $y_0, y_1, y_2$ . In other words, $X$ is a complete intersection in $\PP^3 \times \PP^2$ of type $(1,1) \cdot (2,1)$ ; we have $-K_X = (1,1)$ .
Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+m)!(2l+m)!}{l! l!l!l! m! m! m!}.$
Regularizing this gives period sequence 62:
$1+22 t^2+174 t^3+2514 t^4+34200 t^5+\cdots$
the blow up of $V_4$ with centre an elliptic curve which is the intersection of two hyperplane sections. This is a complete intersection in a toric variety $F$ . The divisor diagram for $F$ is $\begin{array}{l l l l l l l} 1 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 \end{array}$ .
Note that $F$ is a scroll over $\PP^1$ with fibre $\PP^4$ . There is a morphism $F \to \PP^4$ , which is the blow up along $x_0=x_1=0$ ; this map sends $[s_0,s_1,x, x_2, x_3, x_4,x_5]$ to $[s_0x,s_1x,x_2,x_3,x_4,x_5]$ . There are two line bundles $L, M$ on $F$ : $s_0,s_1$ are section of $L$ ; $xs_0, xs_1,x_2, x_3, x_4, x_5$ are sections of $M$ . $X$ is a complete intersection of divisors $2M$ and $2M$ in $F$ ; we have $-K_X=L+M$ . Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(2m)!(2m)!}{l! l! m! m! m! m!\Gamma (1+m-l)}.$
Regularizing this gives period sequence 40:
$1+28 t^2+216 t^3+3516 t^4+49680 t^5+\cdots$
the blow up of $V_3$ with centre a line on it. This is a hypersurface in a toric variety $F$ . The divisor diagram for $F$ is $\begin{array}{llllll} 1 & 1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{array}$ .
Note that $F$ is a scroll over $\PP^2$ with fibre $\PP^2$ . There is a morphism $F \to \PP^4$ , which is the blow up along $x_0=x_1=x_2=0$ ; this map sends $[s_0,s_1,s_2, x, x_3, x_4]$ to $[s_0x,s_1x,s_2x,x_3,x_4]$ . There are two line bundles $L, M$ on $F$ : $s_0,s_1$ are section of $L$ ; $xs_0, xs_1,xs_2, x_3, x_4$ are sections of $M$ . $X$ is cut out by a section of $L+2M$ in $F$ ; we have $-K_X=L+M$ . Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+2m)!}{l! l! l! m! m!\Gamma (1+m-l)}.$
Regularizing this gives period sequence 56:
$1+14 t^2+108 t^3+1074 t^4+13440 t^5+\cdots$
the blow up of $\PP^3$ in a curve $\Gamma$ of degree 6 and genus 3. $\Gamma$ is cut by the equations:
$\begin{pmatrix} l_{00} & l_{01} & l_{02} & l_{03} \\ l_{10} & l_{11} & l_{12} & l_{13} \\ l_{20} & l_{21} & l_{22} & l_{23} \end{pmatrix}$
where the $l_{ij}$ are linear forms. Let $y_0,\ldots,y_3$ be the $3 \times 3$ minors. The relations (szyzgies) between these equations are generated by:
$\begin{cases} l_{00} y_0 + l_{01} y_1 + l_{02} y_2 + l_{03} y_3 = 0 \\ l_{10} y_0 + l_{11} y_1 + l_{12} y_2 + l_{13} y_3 = 0 \\l_{20} y_0 + l_{21} y_1 + l_{22} y_2 + l_{23} y_3 = 0 \end{cases}$
Thus $X$ is given by these three equations in $\PP^3 \times \PP^3$ , where the first factor has co-ordinates $x_0, x_1, x_2, x_3$ and the second factor has co-ordinates $y_0, y_1, y_2,y_3$ . In other words, $X$ is a complete intersection in $\PP^3 \times \PP^3$ of type $(1,1) \cdot (1,1) \cdot(1,1)$ ; we have $-K_X = (1,1)$ .
Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+m)!(l+m)!(l+m)!}{l! l!l!l!m! m! m! m!}.$
Regularizing this gives period sequence 13:
$1+14 t^2+72 t^3+882 t^4+8400 t^5+\cdots$
which is a D3 form (because this is obviously a G-Fano: it is Galkin’s $Y_{20}$ ).
the blow-up of a 3-dimensional quadric $Q$ in a curve $\Gamma$ of genus 2 and degree 6. This is a complete intersection in a toric variety. We have $\Gamma = \PP(1,1,3) \cap Q$ where the embedding $\PP(1,1,3) \hookrightarrow \PP^4$ sends $[s_0:s_1:y] \in \PP(1,1,3)$ to $[s_0^3:s_0^2 s_1:s_0 s_1^2:s_1^3:y] \in \PP^4$ . We blow up $\PP(1,1,3)$ inside $\PP^4$ . The equations defining $\PP(1,1,3)$ are
$\rk \begin{pmatrix} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \end{pmatrix} < 2$
where $[x_0:x_1:x_2:x_3:x_4]$ are co-ordinates on $\PP^4$ . The blown-up variety $F$ is the complete intersection in $\PP^4 \times \PP^2$ cut out by the equations:
$\begin{cases} x_0 y_0 - x_1 y_1 + x_2 y_2 = 0 \\ x_1 y_0 - x_2 y_1 + x_3 y_2 = 0 \end{cases}$
where $y_0, y_1, y_2$ are co-ordinates on $\PP^2$ . Our Fano $X$ is the complete intersection of $F$ with a quadric $q(x_0,x_1,x_2,x_3,x_4)$ . Thus $X$ is a complete intersection of type $(L+M)\cdot(L+M)\cdot(2L)$ in $\PP^4 \times \PP^2$ ; here $L$ is the tautological bundle on $\PP^4$ and $M$ is the tautological bundle on $\PP^2$ .
We have $-K_X = L+N$ and Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+m)!(l+m)!(2l)!}{l! l!l!l!l! m! m! m!}.$
Regularizing this gives period sequence 52:
$1+14 t^2+84 t^3+930 t^4+9720 t^5+\cdots$
the blow-up…
the blow-up of $\PP^3$ with center the intersection of a quadric and a cubic. This is a hypersurface in a scroll $F$ . The divisor diagram for $F$ is
$\begin{array}{llllll} 1 & 1 & 1 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{array}$ .
The projection $F \to \PP^3$ sends $[x_0,x_1,x_2, x_3, s, t]$ to $[x_0,x_1,x_2,x_3]$ . There are two line bundles $L, M$ on $F$ : $x_0,x_1,x_2,x_3$ are section of $L$ ; $s, tx_i$ are sections of $M$ . $X$ is cut out of $F$ by a section of $2L+M$ ; we have $-K_X=L+M$ . Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(2l+m)!}{l! l! l! l! m!\Gamma (1+m-l)}.$
Regularizing this gives period sequence 35:
$1+12 t^2+36 t^3+564 t^4+3600 t^5+\cdots$
the blow-up of $V_4 \subset \PP^5$ with center a conic on it. This is a complete intersection in a toric variety. We give full details of the construction, as it is a model for several other examples (being a blow-up of a projective hypersurface with center a complete intersection in the ambient projective space).We begin by constructing the blow-up $Y$ of $\PP^5$ with center the conic $x_0=x_1=x_2=q=0$ where $q$ is a quadratic polynomial in $x_0,\ldots,x_5$ . To do this, introduce new co-ordinates $s_0, s_1, s_2, t, x$ and impose the relation:
$\left(\begin{array}{c} x_0 \\ x_1 \\ x_2 \\ q \end{array}\right) = x \left(\begin{array}{c} s_0 \\ s_1 \\ s_2 \\ t \end{array}\right).$
Thus we construct $Y$ as a hypersurface in the toric variety $F$ with divisor diagram $\begin{array}{llllllll} 1 & 1 & 1 & -1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \end{array}$ . The co-ordinates on $F$ here are $s_0,s_1,s_2,x,x_3,x_4,x_5,t$ ; the equation defining $Y$ is $xt = q(s_0x,s_1x,s_2x,x_3,x_4,x_5)$ . The map from $Y$ to $\PP^5$ sends $[s_0,s_1,s_2,x,x_3,x_4,x_5]$ to $[s_0 x, s_1 x, s_2 x, x_3, x_4, x_5]$ . It is easy to check that this is the blow-up of $\PP^5$ in the conic $x_0=x_1=x_2=q=0$ . Introduce line bundles $L, M$ such that $s_0, s_1, s_2$ are sections of $L$ and $s_0 x, s_1 x, s_2 x, x_3, x_4, x_5$ are sections of $M$ ; note that $Y$ is cut out of $F$ by a section of $2M$ . The Fano $X$ is a complete intersection of 3 divisors $L+M$ , $L+M$ , and $2M$ on $F$ . We have $-K_X = L+M$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+m)!(l+m)!(2m)!}{l! l! l! \Gamma (1+m-l) m! m! m! (l+m)!}.$
Regularizing this gives period sequence 59:
$1+10 t^2+60 t^3+510 t^4+4920 t^5+\cdots$
the blow-up…
the double cover of $\PP^1 \times \PP^2$ with branch locus a divisor of bidegree $(2,2)$ . This is a hypersurface in a toric variety $F$ with divisor diagram $\begin{array}{l l l l l l} 1 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 1 & 1 & 1 \end{array}$ . With coordinates $x_0, x_1,y_0,y_1,y_2 ,z$ , the defining equation of $X$ is $z^2=f_{2,2}(x_0,x_1;y_0,y_1,y_2)$ . Denote by $L$ the line bundle with sections $x_i$ and by $M$ the line bundle with sections $y_j$ ; then $-K_X = L+2M$ .Quantum Lefschetz gives
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{(2l+2m)!}{l! l! m! m! m! (l+m)!}.$
Regularizing this gives period sequence 60:
$1+6 t^2+48 t^3+282 t^4+2400 t^5+\cdots$
the blow-up of $V_4$ with center a line on it. We proceed as in example 16; $X$ here is a complete intersection in the toric variety $F$ with divisor diagram
$\begin{array}{lllllll} 1 & 1 & 1 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \end{array}.$
The co-ordinates on $F$ here are $s_0,s_1,s_2,s_3,x,x_4,x_5,t$ ; the map from $Y$ to $\PP^5$ sends $[s_0,s_1,s_2,s_3,x,x_4,x_5]$ to $[s_0 x, s_1 x, s_2 x, s_3 x, x_4, x_5]$ . Introduce line bundles $L, M$ such that $s_0, s_1, s_2, s_3$ are sections of $L$ and $s_0 x, s_1 x, s_2 x, s_3 x, x_4, x_5$ are sections of $M$ . Then $X$ is a complete intersection of 2 divisors $L+M$ , $L+M$ on $F$ . We have $-K_X = L+M$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+m}\frac{(l+m)!(l+m)!}{l! l! l! l! \Gamma (1+m-l) m! m!}.$
Regularizing this gives period sequence 55:
$1+8 t^2+30 t^3+240 t^4+1920 t^5+\cdots$
the blow-up…
the blow-up…
the blow-up…
the blow-up of a quadric with center an intersection of $A \in |\cO(1)|$ and $B \in |\cO(2)|$ . $X$ is a complete intersection of type $(L+2M)\cdot(2M)$ in the toric variety with weight data $\begin{array}{llllllll} x_0 & x_1 & x_2 & x_3 & x_4 & s & t\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & L \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & M \end{array}$ . We have $-K_X = L+2M$ .Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{(l+2m)!(2m)!}{m! m! m! m! m! l! (l+m)!}.$
Regularizing this gives period sequence 29:
$1+8 t^2+12 t^3+216 t^4+720 t^5+\cdots$
A divisor of bidegree $(1,2)$ on $\PP^2 \times \PP^2$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{2l+m}\frac{(l+2m)!}{m! m! m! l! l! l!}.$
Regularizing this gives period sequence 66:
$1+4 t^2+24 t^3+132 t^4+780 t^5+\cdots$
The blow up of $\PP^3$ with centre an elliptic curve which is the complete intersection of two quadrics. $X$ is a divisor of bidegree $(1,2)$ in $\PP^1 \times \PP^3$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{(l+2m)!}{l! l! m! m! m! m!}.$
Regularizing this gives period sequence 28:
$1+4 t^2+24 t^3+60 t^4+720 t^5+\cdots$
the blow up…
the blow up of $\PP^3$ with center a twisted cubic. The twisted cubic in $\PP^3$ with co-ordinates $x_0, x_1, x_2, x_3$ is given by the condition
$\rk \left( \begin{array}{lll} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3\end{array} \right) < 2.$
Let $q_0= x_1 x_3 - x_2^2, q_1 = x_1 x_2-x_0 x_3, q_2 = x_0 x_2 - x_1^2$ . The relations (szyzgies) between these equations are generated by:
$\begin{cases} x_0 q_0 + x_1 q_1 + x_2 q_2 = 0 \\ x_1 q_0 + x_2 q_1 + x_3 q_2 = 0 \end{cases}$
Thus $X$ is given by these two equations in $\PP^3 \times \PP^2$ , where the first factor has co-ordinates $x_0, x_1, x_2, x_3$ and the second factor has co-ordinates $q_0, q_1, q_2$ . In other words, $X$ is a complete intersection in $\PP^3 \times \PP^2$ of type $(1,1) \cdot (1,1)$ .
Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{2l+m}\frac{(l+m)!(l+m)!}{l! l!l!l! m! m! m!}.$
Regularizing this gives period sequence 61:
$1+2 t^2+18 t^3+30 t^4+240 t^5+\cdots$
the blow-up of $\PP^3$ with centre a plane cubic. $X$ is a hypersurface of type $(L+3M)$ in the toric variety with weight data
$\begin{array}{lllllll} x_0 & x_1 & x_2 & x_3 & s & t\\ 0 & 0 & 0 & 0 & 1 & 1 & L \\ 1 & 1 & 1 & 1 & 0 & 2 & M \end{array}$ .
We have $-K_X = L+3M$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+3m}\frac{(l+3m)!}{m! m! m! m! l! (l+2m)!}.$
Regularizing this gives period sequence 33:
$1+18 t^3+24 t^4 + 0t^5 +\cdots$
the blow-up of a quadric 3-fold $Q$ with centre a conic on it. $X$ is a hypersurface of type $2M$ in the toric variety with weight data
$\begin{array}{lllllll} s_0 & s_1 & x& x_2 & x_3 & x_4\\ 1 & 1 & -1 & 0 & 0 & 0 & L \\ 0 & 0 & 1 & 1 & 1 & 1 & M \end{array}$ .
We have $-K_X = L+2M$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{(2m)!}{l! l! \Gamma(1+m-l) m! m! m!}.$
Regularizing this gives period sequence 42:
$1+4 t^2+12 t^3+36 t^4+360 t^5+ \cdots$
the blow-up of $\PP^3$ with center a conic. $X$ is a hypersurface of type $L+M$ in the toric variety with weight data
$\begin{array}{lllllll} x_0 & x_1 & x_2 & x_3 & s & t\\ 0 & 0 & 0 & 0 & 1 & 1 & L \\ 1 & 1 & 1 & 1 & -1 & 0 & M \end{array}$ .
We have $-K_X = L+2M$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{(l+m)!}{m! m! m! m! \Gamma(1+l-m) l!}.$
Regularizing this gives period sequence 70:
$1+12 t^3+24 t^4 + 0 t^5+ \cdots$
the blow-up of a quadric 3-fold $Q$ with center a line on it. $X$ is a hypersurface of type $L+M$ in the toric variety with weight data
$\begin{array}{lllllll} s_0 & s_1 & s_2 & x & x_3 & x_4\\ 1 & 1 & 1 & -1 & 0 & 0 & L \\ 0 & 0 & 0 & 1 & 1 & 1 & M \end{array}$ .
We have $-K_X = L+2M$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{(l+m)!}{l! l! l! \Gamma(1+m-l) m! m!}.$
Regularizing this gives period sequence 48:
$1+2 t^2+12 t^3+6 t^4+180 t^5+ \cdots$
a divisor on $\PP^2 \times \PP^2$ of bidegree $(1,1)$ . Quantum Lefschetz gives:
$I_X = \sum_{l,m\geq 0} t^{2l+2m}\frac{(l+m)!}{l! l! l! m! m! m!}.$
Regularizing this gives period sequence 6:
$1+4 t^2+60 t^4 + 0 t^5+ \cdots$
Note that this is a D3 form.
the blow-up of $\PP^3$ with center a line. $X$ is a toric variety with weight data:
$\begin{array}{llllll} s_0 & s_1 & x & x_2 & x_3 \\ 1 & 1 &- 1 & 0 & 0 & L \\ 0 & 0 & 1 & 1 & 1 & M \end{array}$ .
We have $-K_X = L+3M$ and:
$I_X = \sum_{l,m\geq 0} t^{l+3m}\frac{1}{l! l! \Gamma(1+m-l) m! m!}.$
Regularizing this gives period sequence 54:
$1+6 t^3+24 t^4 + 0 t^5+ \cdots$
$X = \PP^1 \times \PP^2$ . We have:
$I_X = \sum_{l,m\geq 0} t^{2l+3m}\frac{1}{l! l! m! m! m!}.$
Regularizing this gives period sequence 44:
$1+2 t^2+6 t^3+6 t^4+120 t^5 + \cdots$
$V_7$ , which is the blow-up of $\PP^3$ at a point. This is a toric variety with weight data:
$\begin{array}{llllll} s_0 & s_1 & s_2 & x & x_3 \\ 1 & 1 & 1 & -1 & 0 & L \\ 0 & 0 & 0 & 1 & 1 & M \end{array}$ .
We have $-K_X = 2L+2M$ and:
$I_X = \sum_{l,m\geq 0} t^{2l+2m}\frac{1}{l! l! l! \Gamma(1+m-l) m!}.$
Regularizing this gives period sequence 30:
$1+2 t^2+30 t^4+ 0 t^5+ \cdots$
the scroll $\PP(\cO \oplus \cO(2))$ over $\PP^2$ . This is a toric variety with weight data:
$\begin{array}{llllll} x_0 & x_1 & x_2 & s & t \\ 1 & 1 & 1 & -2 & 0 & L \\ 0 & 0 & 0 & 1 & 1 & M \end{array}$ .
We have $-K_X = L+2M$ and:
$I_X = \sum_{l,m\geq 0} t^{l+2m}\frac{1}{l! l! l! \Gamma(1+m-2l) m!}.$
Regularizing this gives period sequence 58:
$1+2 t^2+6 t^4+60 t^5+ \cdots$

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2 Comments

Sergey says:

September 7, 2010 at 6:12 am

14. This variety X is a section of half-anticanonical class on $P^1 \times V_5$ ,
where $V = V_5$ is del Pezzo threefold of degree 5.

Regularized I-series for V is $1 + 6 t^2 + 114 t^4 + 2940 t^6 + 87570 t^8 + 2835756 t^{10} + \cdots$
Nonregularized $I_V = 1 + 3 t^2 + 19/4 t^4 + 49/12 t^6 + 139/64 t^8 + 3751/4800 t^{10} +$
For line $I_{P^1} = \sum_n \t^{2n} \frac{1}{n!^2}$

For product fourfold we have
$I_{V \times P^1} = I_V \cdot I_{P^1} = 1 + 4 t^2 + 8 t^4 + 173/18 t^6 + 271/36 t^8 + 14801/3600 t^{10} +$

So for threefold section X we have to change t^2 to t and do Laplace transform once (for non-regularized):
$I_X = 1 + 4 t + 16 t^2 + 173/3 t^3 + 542/3 t^4 + 14801/30 t^5 + \cdots$

Regularizing gives
$1 + 4 t + 32 t^2 + 346 t^3 + 4336 t^4 + 59204 t^5 + \cdots$

After normalization this gives period sequence 39:
$1 + 16 t^2 + 90 t^3 + 1104 t^4 + 11460 t^5 + 133990 t^6 + 1588860 t^7 + 19463920 t^8 + 242996040 t^9 + 3085849116 t^{10} + \cdots$
Sergey says:

October 1, 2010 at 8:29 am

On 17. This is section of half-anticanonical class on $\PP_{\PP^3} E$ , where $E$ is the null-correlation bundle on $\P^3$ .

References are

Wisniewski (1989b)
Ruled Fano 4-folds of index 2

Szurek-Wisniewski (1990b)
Fano bundles over $\PP^3$ and $Q^3$.

Szurek-Wisniewski (1990c)
Fano bundles of rank 2 on $\PP^3$ and $Q_3$.

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