Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-304-66-a

·

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

Defining data
Ambient
ℙ4P4 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(w1 + w5) + E^(w4 + w5) + E^(2w1)
Bundle
O_1(1) box O_2(1) + Q_1 box O_2(1) + O_1(2)
ZL62 label
40600.wOw3A
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;7) \\ \det\mathcal{E} &\cong \mathcal{O}(4;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 66
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 304
$\chi(X, \mathrm{T}_X)$ = 1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 304
$\mathrm{c}_1^2\mathrm{c}_2$ = 172
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 6 0 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 + x1 - 1 0 x0 - 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 54 0 0 x1 - 29 83
$\mathrm{HH}^{ 4 }$ = 66 0 0 0 0 66
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 61t + 180t^{2} + 61t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(112)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(159)

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( 2t - 1 \right) \left( 2t + 1 \right) \left( 496t^{6} + 8224t^{5} + 3696t^{4} + 568t^{3} - 4t^{2} - 8t - 1 \right) \left( 118604033744896t^{13} + 192440543207424t^{12} + 153963573566208t^{11} + 91273918133888t^{10} + 47096301846960t^{9} + 16568166104128t^{8} + 3134227768656t^{7} + 364909120560t^{6} + 70243475184t^{5} + 15243484616t^{4} + 1382527176t^{3} + 56107972t^{2} - 131923t + 16644 \right)D^{6} + \left( 4000276850147852288t^{21} + 57593214501960810496t^{20} + 109682678307395321856t^{19} + 103559783581290987520t^{18} + 66888421373039014912t^{17} + 35080457635062421504t^{16} + 12871643041458558720t^{15} + 1494357647082216448t^{14} - 962126132413728000t^{13} - 483514901254308992t^{12} - 107060088807046656t^{11} - 20676261905483264t^{10} - 5386408992160592t^{9} - 1175125461304320t^{8} - 156581658119040t^{7} - 12892423707648t^{6} - 1000335441040t^{5} - 102673696296t^{4} - 7012499880t^{3} - 225487272t^{2} + 662073t - 33288 \right)D^{5} + \left( 27060696339235471360t^{21} + 310837186105518325760t^{20} + 578384364248839929856t^{19} + 564026002929355825152t^{18} + 399193997579704936448t^{17} + 236676645802994612736t^{16} + 101721369103698025920t^{15} + 24262197473015283456t^{14} + 1729771540475530688t^{13} - 278459813168438720t^{12} - 12186694695598848t^{11} - 4357623469790688t^{10} - 8376388570332016t^{9} - 2360009188975440t^{8} - 311284854353308t^{7} - 23331002770352t^{6} - 610287234848t^{5} + 73085308128t^{4} + 6419797836t^{3} + 250114016t^{2} + 270295t + 16644 \right)D^{4} + t \left( 92947609165200097280t^{20} + 864003034556200386560t^{19} + 1566907884905162686464t^{18} + 1552645694885633589248t^{17} + 1155174074187809340416t^{16} + 726815987724409553408t^{15} + 328882887721289370752t^{14} + 88460535943140945920t^{13} + 13214478238339270272t^{12} + 1924614922407596032t^{11} + 566138507505436416t^{10} + 81938992593288128t^{9} - 9752742534397424t^{8} - 4416498934579200t^{7} - 592556755616472t^{6} - 42442792621504t^{5} - 1403474584976t^{4} - 17137749624t^{3} - 895210064t^{2} - 8628696t - 1533 \right)D^{3} + 4 t ^{2} \left( 42591182933927133184t^{19} + 326793590001308270592t^{18} + 576441991641486950400t^{17} + 577095590343094902784t^{16} + 444927413807847855360t^{15} + 290868024977354633728t^{14} + 134619682374820309904t^{13} + 37398099672453614656t^{12} + 6459222062625189840t^{11} + 1237403184906575824t^{10} + 346198470863847840t^{9} + 57088309352062952t^{8} + 1345534794177776t^{7} - 807607700092500t^{6} - 128959958213325t^{5} - 9814529995032t^{4} - 321826963176t^{3} - 5398183616t^{2} - 275462771t - 1707112 \right)D^{2} + 8 t ^{2} \left( 19648418646314450944t^{19} + 127565525130926358528t^{18} + 218672385599080144896t^{17} + 220738064586366173184t^{16} + 175255666624728321024t^{15} + 117778691272756270848t^{14} + 55062575372485050800t^{13} + 15325516286087731008t^{12} + 2745494284262890128t^{11} + 579508184764260080t^{10} + 166756449442670256t^{9} + 29411687578413624t^{8} + 1964238308506352t^{7} - 84105311412012t^{6} - 27644732656791t^{5} - 2451014440648t^{4} - 83974727058t^{3} - 2009261184t^{2} - 87065300t - 366168 \right)D + 48 t ^{2} \left( 1176552014749368320t^{19} + 6663031326313021440t^{18} + 11114306560769949696t^{17} + 11300252050291851264t^{16} + 9197792137939572480t^{15} + 6308474833021129728t^{14} + 2960942817629057760t^{13} + 816301843227424768t^{12} + 146278527829664704t^{11} + 32678522511954272t^{10} + 9815569923616816t^{9} + 1810702208180816t^{8} + 158362550863296t^{7} + 4009390821544t^{6} - 700512612742t^{5} - 85655802920t^{4} - 3229349806t^{3} - 105228836t^{2} - 3762055t - 11096 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ6P6

|λ| = 8.246929 λ ≈ 8.246929; multiplicity 1; Jordan blocks 1 λ ≈ -4.319438 - 5.924335i; multiplicity 1; Jordan blocks 1 λ ≈ -4.319438 + 5.924335i; multiplicity 1; Jordan blocks 1 λ ≈ -3.773015 - 1.950202i; multiplicity 1; Jordan blocks 1 λ ≈ -3.773015 + 1.950202i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ 2; multiplicity 1; Jordan blocks 1 λ ≈ -0.06202367; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$8.24692911$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.246929$11
$-4.319438 - 5.924335\mathrm{i}$11
$-4.319438 + 5.924335\mathrm{i}$11
$-3.773015 - 1.950202\mathrm{i}$11
$-3.773015 + 1.950202\mathrm{i}$11
$-2$11
$2$11
$-0.06202367$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{6} + 8 \lambda^{5} + 4 \lambda^{4} - 568 \lambda^{3} - 3696 \lambda^{2} - 8224 \lambda - 496\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{6} + 8 \lambda^{5} + 4 \lambda^{4} - 568 \lambda^{3} - 3696 \lambda^{2} - 8224 \lambda - 496\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 12 & 6 & 12 & 0 & 0 & 0 \\ 1 & -2 & 0 & 2 & 4 & 4 & 12 & 12 & 0 \\ 2 & 2 & 0 & 0 & -4 & 0 & 12 & 6 & 0 \\ 0 & 2 & 1 & -2 & 0 & 0 & 0 & 4 & 12 \\ 0 & 1 & 0 & 0 & -2 & 0 & 4 & 2 & 0 \\ 0 & 0 & 2 & 2 & 2 & 0 & 0 & -4 & 0 \\ 0 & 0 & 0 & 2 & -\frac{1}{2} & 3 & 0 & 2 & -2 \\ 0 & 0 & 0 & 1 & 3 & 0 & 0 & -2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$