Rank 3 Fano 3-folds

  1. 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.
  2. 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_jConsider 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.

  3. 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
  4. 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
  5. 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^2F 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
  6. 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
  7. 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.

  8. 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).
  9. 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
  10. 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
  11. 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.

  12. 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.

  13. the blow-up…
  14. 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
  15. 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 QX 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).
  16. 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
  17. 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).
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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

  25. 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
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. 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.

5 Comments

  1. Sergey says:

    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}

  2. Sergey says:

    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).

  3. Sergey says:

    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.

  4. Sergey says:

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

  5. Sergey says:

    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.

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