Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-192-45-d

·

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

Defining data
Ambient
ℙ3P3 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(2w4) + E^(w2 + w4)
Bundle
O_2(2) + Q_1^v(1) box O_2(1)
ZL62 label
3050.Bw8
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 192
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 192
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 0
Hodge diamond
1
00
0 2 0
0 2 2 0
0 0 2 0 0
0 2 2 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 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 50 0 0 x1 + 14 36
$\mathrm{HH}^{ 4 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(238)
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 12t - 1 \right) \left( 4t + 1 \right) ^{3} \left( 64t^{2} + 1 \right) \left( 4017881088t^{7} + 2930012160t^{6} + 857934080t^{5} + 127169920t^{4} + 10396080t^{3} + 797992t^{2} + 1745t + 60 \right)D^{6} + \left( 4t + 1 \right) ^{2} \left( 160458099130368t^{11} + 126013962977280t^{10} + 42255204483072t^{9} + 7715879518208t^{8} + 843696197632t^{7} + 80231698432t^{6} + 5558512640t^{5} + 618142208t^{4} + 33406592t^{3} + 3175568t^{2} + 4755t + 120 \right)D^{5} + \left( 4t + 1 \right) \left( 3320248358928384t^{12} + 3189135409938432t^{11} + 1332416155484160t^{10} + 312853711683584t^{9} + 45389357932544t^{8} + 4654456420352t^{7} + 300583454208t^{6} + 1167263360t^{5} - 989633504t^{4} - 43987544t^{3} - 2814016t^{2} - 4275t - 60 \right)D^{4} + t \left( 35053923194634240t^{12} + 40730709026930688t^{11} + 20845370023084032t^{10} + 6129468861841408t^{9} + 1140205352615936t^{8} + 146338753560576t^{7} + 13737312387072t^{6} + 796926263808t^{5} + 17514259712t^{4} + 158989824t^{3} + 193984t^{2} + 54280t + 65 \right)D^{3} + 8 t ^{2} \left( 6239351470030848t^{11} + 7059575352066048t^{10} + 3541916502982656t^{9} + 1024342695936000t^{8} + 187648778118656t^{7} + 23998322639104t^{6} + 2217591495072t^{5} + 115231269744t^{4} + 1739828870t^{3} + 18768504t^{2} + 506743t + 5880 \right)D^{2} + 16 t ^{2} \left( 2277270714580992t^{11} + 2533600986660864t^{10} + 1258818737799168t^{9} + 361666702245888t^{8} + 65857133320448t^{7} + 8451824283008t^{6} + 770887362416t^{5} + 36044118472t^{4} + 335274697t^{3} + 4242112t^{2} + 156240t + 1320 \right)D + 32 t ^{2} \left( 333259128963072t^{11} + 366848322306048t^{10} + 181594982449152t^{9} + 52137511256064t^{8} + 9489661629184t^{7} + 1226287736448t^{6} + 110950119632t^{5} + 4772407608t^{4} + 26861331t^{3} + 462946t^{2} + 14645t + 120 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ5P5

|λ| = 12 λ ≈ 12; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 8i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 8i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 3; Jordan blocks 1, 2 λ ≈ 0; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$12$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12$11
$0 - 8\mathrm{i}$11
$0 + 8\mathrm{i}$11
$-4$31, 2
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right)^{3} \left(\lambda - 12\right) \left(\lambda^{2} + 64\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right)^{2} \left(\lambda - 12\right) \left(\lambda^{2} + 64\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 4 & 12 & 12 & 96 & 80 & 160 \\ 2 & 0 & 4 & 4 & 0 & 0 & 0 & 32 \\ 1 & 0 & 0 & 2 & 2 & 24 & 24 & 64 \\ 0 & 1 & 4 & 0 & 0 & 4 & 4 & 24 \\ 0 & 2 & -2 & 4 & 0 & 4 & 0 & -24 \\ 0 & 0 & 0 & 1 & -1 & -4 & -2 & 0 \\ 0 & 0 & 0 & 2 & 3 & 8 & 4 & 4 \\ 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$