Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-384-81-b

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
ℙ2P2 × ℙ2P2 × Fl(2, 3, 5)
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_5node 6: fundamental weight ω_6 (marked)node 7: fundamental weight ω_7 (marked)node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w4 + w5) + E^(w2 + w8)
Bundle
Q_2 box U_3,3^v + Q_1 box Q_3,1^v ot U_3,2^v ot wedge^2 U_3,3^v
ZL62 label
202045.wLw9
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 81
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 384
$\chi(X, \mathrm{T}_X)$ = 11

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 384
$\mathrm{c}_1^2\mathrm{c}_2$ = 192
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 11 0 x0 - 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 59 0 0 x1 - 52 111
$\mathrm{HH}^{ 4 }$ = 81 0 0 0 0 81
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 11t + 11t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(257)
Coates et al. (2020) P1 x MM(2,32)
Coates–Kasprzyk–Prince (2015) CKP(325)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 8t^{2} - 4t + 1 \right) \left( 8t^{2} + 4t + 1 \right) \left( 28t^{2} - 4t - 1 \right) \left( 28t^{2} + 4t - 1 \right) \left( 1245440t^{6} + 159472t^{4} - 353t^{2} + 2 \right)D^{6} + 2 \left( 687403171840t^{14} + 55847092224t^{12} - 2940634112t^{10} + 646645248t^{8} - 7816080t^{6} - 602296t^{4} + 771t^{2} - 4 \right)D^{5} + 4 \left( 2968331878400t^{14} + 342433251328t^{12} - 16569335296t^{10} + 962246144t^{8} + 20691276t^{6} + 609636t^{4} - 327t^{2} + 2 \right)D^{4} + 8 t ^{2} \left( 6405347737600t^{12} + 905030160384t^{10} - 30840349696t^{8} + 529133120t^{6} + 4111656t^{4} - 69016t^{2} - 79 \right)D^{3} + 64 t ^{2} \left( 1805409751040t^{12} + 290148947200t^{10} - 6893123664t^{8} + 13678020t^{6} - 1333787t^{4} - 13834t^{2} - 3 \right)D^{2} + 256 t ^{4} \left( 497488517120t^{10} + 86925519744t^{8} - 1508603672t^{6} - 6132868t^{4} - 554137t^{2} - 2494 \right)D + 1536 t ^{4} \left( 34174873600t^{10} + 6291237120t^{8} - 85724560t^{6} - 530810t^{4} - 38831t^{2} - 116 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × Fl(2, 3, 5)

|λ| = 7.656854 λ ≈ -7.656854; multiplicity 1; Jordan blocks 1 λ ≈ 7.656854; multiplicity 1; Jordan blocks 1 λ ≈ -3.656854; multiplicity 1; Jordan blocks 1 λ ≈ 3.656854; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2 + 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2 - 2i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2 + 2i; multiplicity 2; Jordan blocks 1, 1 2 2 2 2
Sector dimension
12
Spectral radius
$7.65685425$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-7.656854$11
$7.656854$11
$-3.656854$11
$3.656854$11
$-2 - 2\mathrm{i}$21, 1
$-2 + 2\mathrm{i}$21, 1
$2 - 2\mathrm{i}$21, 1
$2 + 2\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 4 \lambda + 8\right)^{2} \left(\lambda^{2} + 4 \lambda - 28\right) \left(\lambda^{2} - 4 \lambda + 8\right)^{2} \left(\lambda^{2} - 4 \lambda - 28\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{2} + 4 \lambda + 8\right) \left(\lambda^{2} + 4 \lambda - 28\right) \left(\lambda^{2} - 4 \lambda + 8\right) \left(\lambda^{2} - 4 \lambda - 28\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 4 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 \\ 2 & 0 & 0 & 0 & 2 & 4 & 2 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & -4 \\ 2 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 4 \\ 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 \\ 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 4 & 0 \\ 0 & 0 & 2 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 & 4 & 2 & 6 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 4 & -4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$