Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-69-21-a

·

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

Defining data
Ambient
Fl(1, 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_4ω1ω2ω3ω4
Bundle
E^(w3) + E^(w4) + E^(2w1 + w3)
Bundle
O(0,1) + Q_1^v ot wedge^2 U_2^v ot U_3^v + O(2,1)
ZL62 label
44.32x317
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 69
$\chi(X, \mathrm{T}_X)$ = -40

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 69
$\mathrm{c}_1^2\mathrm{c}_2$ = 102
$\mathrm{c}_2^2$ = 151
$\mathrm{c}_1\mathrm{c}_3$ = -4
$\mathrm{c}_4$ = 68
Hodge diamond
1
00
0 2 0
0 0 0 0
0 5 52 5 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 + 40 0 x0 + 40 x2
$\mathrm{HH}^{ 3 }$ = x1 + 4 0 0 x1 4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 31 0 0 x1 - x2 + 10 0 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 35t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(664)
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) ^{3} \left( 25589t^{6} - 30561t^{5} - 32562t^{4} - 8458t^{3} - 664t^{2} - 2t + 1 \right) \left( 181908839303960t^{12} + 412539629951240t^{11} + 395840664062120t^{10} + 205719036745720t^{9} + 59982946647350t^{8} + 7926471734512t^{7} - 645843341849t^{6} - 462643252372t^{5} - 86779590633t^{4} - 8455486439t^{3} - 405570478t^{2} - 5347831t + 42504 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 293256513203789043720t^{19} + 455096124658427396160t^{18} - 135140293808271125120t^{17} - 886407798616168019680t^{16} - 992373730211936161950t^{15} - 597803469213301864196t^{14} - 220560648545357673631t^{13} - 47760490992587158462t^{12} - 3354266411734129594t^{11} + 1432100165656229484t^{10} + 575467055222702733t^{9} + 111239820114656138t^{8} + 13571236755001733t^{7} + 1079631697206451t^{6} + 54423308004019t^{5} + 1747761558273t^{4} + 51318835664t^{3} + 1755075854t^{2} + 22453924t - 127512 \right)D^{5} + \left( 3t + 1 \right) \left( 7331412830094726093000t^{20} + 14671664168372893145400t^{19} + 5555281479715734455000t^{18} - 11805082317942126990760t^{17} - 18301271586246666169310t^{16} - 12724005697158135057830t^{15} - 5255971952299639556175t^{14} - 1289844874789548862035t^{13} - 124280208892304067012t^{12} + 34517219267900531433t^{11} + 17467550149937795464t^{10} + 3905381865741845887t^{9} + 552441447217378165t^{8} + 51899974098815229t^{7} + 3123368345115599t^{6} + 105619848071229t^{5} + 1120661093185t^{4} - 45084080612t^{3} - 2067101360t^{2} - 17309477t + 85008 \right)D^{4} + t \left( 92375801659193548771800t^{20} + 224074424482514568705600t^{19} + 178372238702986330365000t^{18} - 21946530861851257761680t^{17} - 152422079245094124315490t^{16} - 137108011371779865435200t^{15} - 67132178975198292920435t^{14} - 19736321879852642945140t^{13} - 2850758144197110710791t^{12} + 271973838626213760472t^{11} + 255935968763955489005t^{10} + 70147887599254436264t^{9} + 11775795264193099946t^{8} + 1325651945228960179t^{7} + 99039664290346250t^{6} + 4517418421133600t^{5} + 97274348805483t^{4} - 293488556315t^{3} - 44323576044t^{2} - 444876540t + 581504 \right)D^{3} + 2 t \left( 102053266594918587214560t^{20} + 254665388800895819347260t^{19} + 232268161902812599325700t^{18} + 47681632540720336571820t^{17} - 85539086962168216604940t^{16} - 90931033807654767213148t^{15} - 44783131873243782565295t^{14} - 12428089066956962160260t^{13} - 1427596944811344160380t^{12} + 322185278256224062674t^{11} + 191133870745649910059t^{10} + 46630269476210835932t^{9} + 7144641496144731125t^{8} + 727857149795251740t^{7} + 47631293227465186t^{6} + 1743404239337088t^{5} + 18736296060401t^{4} - 764182319040t^{3} - 23017426348t^{2} - 85511148t + 85008 \right)D^{2} + 4 t ^{2} \left( 55425480995516129263080t^{19} + 141142001794318725130260t^{18} + 138969826043997942554100t^{17} + 50474278548744011849900t^{16} - 19666938393752076794610t^{15} - 30856583540546361835644t^{14} - 15876352715948617134310t^{13} - 4280512660767859106200t^{12} - 402874861364206098703t^{11} + 146749973766964875768t^{10} + 72284342577665365772t^{9} + 16358358425882007522t^{8} + 2336381660272029027t^{7} + 218290893720677177t^{6} + 12544458583296572t^{5} + 349253652334390t^{4} - 1470308373424t^{3} - 325806368511t^{2} - 6561451328t - 6116160 \right)D + 96 t ^{3} \left( 942610221012179069100t^{18} + 2432917169022961719450t^{17} + 2502451164768585470850t^{16} + 1113331429112457073300t^{15} - 67235133146836068225t^{14} - 347070329138680958280t^{13} - 192645291013373053400t^{12} - 51671779267542589610t^{11} - 4218854667671900450t^{10} + 2027987885517488600t^{9} + 911589402205131190t^{8} + 196151356139389905t^{7} + 26513178916324559t^{6} + 2295240720000690t^{5} + 115596631213410t^{4} + 2161254419943t^{3} - 70180052796t^{2} - 4028965506t - 64552950 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 32.01409 λ ≈ 32.01409; multiplicity 1; Jordan blocks 1 λ ≈ -11.97022 - 0.8992957i; multiplicity 1; Jordan blocks 1 λ ≈ -11.97022 + 0.8992957i; multiplicity 1; Jordan blocks 1 λ ≈ -3.58694; multiplicity 1; Jordan blocks 1 λ ≈ -3.001874; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 3; Jordan blocks 3 λ ≈ 0.5151655; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$32.014094$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$32.01409$11
$-11.97022 - 0.8992957\mathrm{i}$11
$-11.97022 + 0.8992957\mathrm{i}$11
$-3.58694$11
$-3.001874$11
$-3$33
$0.5151655$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda^{6} - 2 \lambda^{5} - 664 \lambda^{4} - 8458 \lambda^{3} - 32562 \lambda^{2} - 30561 \lambda + 25589\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda^{6} - 2 \lambda^{5} - 664 \lambda^{4} - 8458 \lambda^{3} - 32562 \lambda^{2} - 30561 \lambda + 25589\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 36 & 32 & 390 & 315 & 126 & 1648 & 656 & 4570 \\ 1 & 0 & 4 & 64 & 56 & 16 & 204 & 84 & 656 \\ 1 & 5 & -1 & 60 & 40 & 24 & 219 & 78 & 664 \\ 0 & 1 & 1 & -1 & 3 & 1 & 36 & 16 & 72 \\ 0 & 1 & 0 & 4 & -2 & 2 & 8 & 0 & 36 \\ 0 & 0 & 2 & 10 & 10 & -2 & 32 & 16 & 102 \\ 0 & 0 & 0 & 5 & 5 & 1 & 5 & 4 & 32 \\ 0 & 0 & 0 & -\frac{5}{2} & -5 & 1 & -\frac{5}{2} & -6 & -12 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$