Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-92-26-a

·

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

Defining data
Ambient
Fl(1, 2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w6) + E^(2w2) + E^(w1 + 2w2)
Bundle
wedge^4 Q_1^v ot U_2^v ot U_3^v + O(0,2) + O(1,2)
ZL62 label
602.57x301
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,6) \\ \det\mathcal{E} &\cong \mathcal{O}(1,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 26
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 92
$\chi(X, \mathrm{T}_X)$ = -47

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 92
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 146
$\mathrm{c}_1\mathrm{c}_3$ = -12
$\mathrm{c}_4$ = 78
Hodge diamond
1
00
0 2 0
0 0 0 0
0 6 60 6 0
0 0 0 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 + x2 + 47 0 x0 + 47 x2
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 15
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 45 0 0 x1 - x2 + 19 0 26
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 21t + 48t^{2} + 21t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(632)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(642)

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( t + 1 \right) ^{3} \left( 7936t^{4} + 8448t^{3} + 480t^{2} - 16t - 1 \right) \left( 1813471232t^{9} + 6878861312t^{8} + 9797574400t^{7} + 5655358464t^{6} + 909198416t^{5} + 65573664t^{4} + 8797074t^{3} + 623946t^{2} - 4690t + 23 \right)D^{6} + 2 \left( t + 1 \right) ^{2} \left( 122329515425792t^{14} + 626876391882752t^{13} + 1320796202172416t^{12} + 1414964637597696t^{11} + 783986199214080t^{10} + 212754899795968t^{9} + 30479143553408t^{8} + 3398133906432t^{7} + 442135538648t^{6} + 32419160872t^{5} + 581381969t^{4} + 15427541t^{3} + 1337061t^{2} - 7518t + 23 \right)D^{5} + \left( t + 1 \right) \left( 1655046385172480t^{15} + 9218739864600576t^{14} + 21568607744753664t^{13} + 26613875946061824t^{12} + 18104485953703936t^{11} + 6646608361111552t^{10} + 1295710380431616t^{9} + 162767654800896t^{8} + 20517405984144t^{7} + 2463848840112t^{6} + 132770684838t^{5} + 41031414t^{4} - 28417036t^{3} - 2760785t^{2} + 6457t - 23 \right)D^{4} + t \left( 5684724540375040t^{15} + 35018347543265280t^{14} + 92319292617457664t^{13} + 132500442141622272t^{12} + 110248133575266304t^{11} + 53174845238112256t^{10} + 14488542448661760t^{9} + 2300507857466368t^{8} + 291461698610832t^{7} + 42216275369936t^{6} + 4416987472094t^{5} + 179243278610t^{4} - 803346088t^{3} + 1762136t^{2} + 247720t + 25 \right)D^{3} + t ^{2} \left( 10419596372738048t^{14} + 61102927706587136t^{13} + 152387259879784448t^{12} + 203862594853502976t^{11} + 154679982654652416t^{10} + 66295977758119936t^{9} + 15783156000590336t^{8} + 2295839266716416t^{7} + 309870385616640t^{6} + 47097916284624t^{5} + 4222808544732t^{4} + 122287329524t^{3} - 1115605936t^{2} + 16979312t + 167233 \right)D^{2} + 16 t ^{2} \left( 600853796356096t^{14} + 3398522753384448t^{13} + 8129879349657600t^{12} + 10291269727014912t^{11} + 7238374749744128t^{10} + 2809182388246784t^{9} + 600059868195712t^{8} + 83936837044640t^{7} + 12359442990428t^{6} + 1849286278734t^{5} + 139707577612t^{4} + 2621033596t^{3} - 30106766t^{2} + 786037t + 3703 \right)D + 48 t ^{2} \left( 71958538485760t^{14} + 396832969064448t^{13} + 921393071259648t^{12} + 1119110172131328t^{11} + 741681953533952t^{10} + 265560836283392t^{9} + 52087010980864t^{8} + 7220914491392t^{7} + 1154874559664t^{6} + 165506676408t^{5} + 10651687396t^{4} + 111199060t^{3} - 1217273t^{2} + 62614t + 184 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 7)

|λ| = 22.56186 λ ≈ 22.56186; multiplicity 1; Jordan blocks 1 λ ≈ -19.24815; multiplicity 1; Jordan blocks 1 λ ≈ -18.31599; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 3 λ ≈ -0.9977166; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1 3
Sector dimension
8
Spectral radius
$22.5618617$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22.56186$11
$-19.24815$11
$-18.31599$11
$-1$33
$-0.9977166$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{3} \left(\lambda^{4} + 16 \lambda^{3} - 480 \lambda^{2} - 8448 \lambda - 7936\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{3} \left(\lambda^{4} + 16 \lambda^{3} - 480 \lambda^{2} - 8448 \lambda - 7936\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 24 & 72 & 36 & 1176 & 776 & 1440 \\ 1 & -1 & 4 & 36 & -20 & 90 & 318 & 576 \\ 1 & 3 & -4 & 48 & 88 & 54 & -210 & 600 \\ 0 & 2 & 1 & -1 & 8 & 60 & -12 & 18 \\ 0 & 0 & 1 & 3 & -8 & 24 & 64 & 54 \\ 0 & 0 & 0 & 2 & 1 & -1 & 4 & 24 \\ 0 & 0 & 0 & 0 & 1 & 3 & -4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$