Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-194-45-a

·

Blow-up of a smooth quadric fourfold along a smooth elliptic scroll of degree $6$.

  • $\pi_1\colon X\to Y$ maps birationally onto a fourfold $Y$ of degree $24$ in the Lagrangian Grassmannian $\mathrm{LG}(3,6)$, with two singular points; it contracts two disjoint planes, each with normal bundle $\mathcal O(-1)^{\oplus 2}$, to these points.

  • $\pi_2\colon X\to Q^4$ is the blow-up of a smooth quadric fourfold (1-512-105-a) along a smooth elliptic scroll of degree $6$, ruled over a plane cubic curve.

Varieties in this family can be blown down to

Defining data
Ambient
Fl(1, 3, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w2) + E^(w4)
Bundle
2 copies of U_2^v ot U_3^v + Q_1^v ot wedge^2 U_2^v ot U_3^v
ZL62 label
54.311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,5) \\ \det\mathcal{E} &\cong \mathcal{O}(2,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 194
$\chi(X, \mathrm{T}_X)$ = -3

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 194
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 6
Hodge diamond
1
00
0 2 0
0 1 1 0
0 0 4 0 0
0 1 1 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 3 0 x0 + 3 x1
$\mathrm{HH}^{ 3 }$ = x1 + 43 0 0 x1 + 6 37
$\mathrm{HH}^{ 4 }$ = 45 0 0 0 0 45
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 112t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 3t + 1 \right) ^{2} \left( t - 1 \right) ^{3} \left( 4t^{2} - 12t + 1 \right) \left( 68t^{2} + 12t + 1 \right) \left( 833767728t^{9} - 358454208t^{8} + 17467416t^{7} - 180501604t^{6} + 509158967t^{5} + 116196589t^{4} - 13158404t^{3} + 8949933t^{2} + 81219t + 1164 \right)D^{6} + \left( 3t + 1 \right) \left( t - 1 \right) ^{2} \left( 14287443787008t^{15} - 44081250396672t^{14} + 23023157168640t^{13} - 6158327823360t^{12} + 19306031625440t^{11} - 27974253028064t^{10} - 2409398082640t^{9} + 4857966498416t^{8} - 30851958275t^{7} + 51093508196t^{6} + 69297255461t^{5} + 4331953882t^{4} - 58234459t^{3} + 35864154t^{2} + 234345t + 2328 \right)D^{5} + \left( t - 1 \right) \left( 357186094675200t^{17} - 1101611299276800t^{16} + 800470143884736t^{15} - 201308923895040t^{14} + 534374763814944t^{13} - 891266032367712t^{12} + 119243374570764t^{11} + 264481606628420t^{10} - 29808627598017t^{9} - 8520941726951t^{8} + 3256627668144t^{7} - 814928608243t^{6} - 231511233517t^{5} - 5626971203t^{4} - 205047536t^{3} + 34262277t^{2} + 256170t + 1164 \right)D^{4} + t \left( 1500181597635840t^{17} - 5339753776200960t^{16} + 6186069140672640t^{15} - 3242831679543168t^{14} + 3361829361082944t^{13} - 5868490303784816t^{12} + 3529872341967504t^{11} + 591292536195988t^{10} - 853054931877599t^{9} + 82148714318209t^{8} + 11555927818930t^{7} - 13814068463420t^{6} + 1662667911788t^{5} + 393670329505t^{4} + 8666170470t^{3} + 184736754t^{2} + 2068779t + 612 \right)D^{3} + t ^{2} \left( 3314686958585856t^{16} - 10467578230124544t^{15} + 10960583719329600t^{14} - 5815692926448384t^{13} + 7686412332265632t^{12} - 12426458980417312t^{11} + 6512325115236092t^{10} + 1284846999174920t^{9} - 1392762849813422t^{8} + 167542235193628t^{7} - 185711225282t^{6} - 23269100744752t^{5} + 2355853940591t^{4} + 366321936494t^{3} + 7067403810t^{2} + 198675870t + 1639203 \right)D^{2} + 6 t ^{2} \left( 600072639054336t^{16} - 1718449468905984t^{15} + 1656267589983552t^{14} - 900003106924224t^{13} + 1414462171823680t^{12} - 2148566190825648t^{11} + 1002349849759764t^{10} + 230652815920056t^{9} - 198741307307096t^{8} + 28459047583230t^{7} - 1256932312202t^{6} - 3351194792294t^{5} + 271857573689t^{4} + 29770094034t^{3} + 624454011t^{2} + 18192552t + 111744 \right)D + 96 t ^{2} \left( 15307975486080t^{16} - 40795343291520t^{15} + 36927398193264t^{14} - 20602028012472t^{13} + 36393794262156t^{12} - 52764415217418t^{11} + 22482031089372t^{10} + 5852066001762t^{9} - 4260722547393t^{8} + 694508187642t^{7} - 36648251079t^{6} - 72544075640t^{5} + 4582123054t^{4} + 372560584t^{3} + 9894420t^{2} + 261624t + 1164 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 6)

|λ| = 11.65685 λ ≈ 11.65685; multiplicity 1; Jordan blocks 1 λ ≈ -6 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -6 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1; multiplicity 3; Jordan blocks 1, 2 λ ≈ 0.3431458; multiplicity 1; Jordan blocks 1 2 3
Sector dimension
9
Spectral radius
$11.6568542$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.65685$11
$-6 - 5.656854\mathrm{i}$11
$-6 + 5.656854\mathrm{i}$11
$-3$21, 1
$1$31, 2
$0.3431458$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda - 1\right)^{3} \left(\lambda^{2} + 12 \lambda + 68\right) \left(\lambda^{2} - 12 \lambda + 4\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda - 1\right)^{2} \left(\lambda^{2} + 12 \lambda + 68\right) \left(\lambda^{2} - 12 \lambda + 4\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 36 & 18 & 27 & 8 & 28 & 20 \\ 1 & 0 & 3 & -6 & 0 & -6 & 9 & 27 & 8 \\ 1 & 0 & -1 & 14 & 8 & 10 & 6 & 21 & 0 \\ 0 & 1 & 1 & -1 & -2 & 0 & -2 & -8 & 6 \\ 0 & 1 & 0 & 3 & 1 & 3 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 2 & -1 & 6 & 20 & 0 \\ 0 & 0 & 0 & 16 & 18 & 4 & 7 & 28 & -8 \\ 0 & 0 & 0 & -2 & -4 & 1 & -2 & -8 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 8 & 0\end{pmatrix}\notag\end{equation}$$