Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-147-36-c

·

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

Defining data
Ambient
Fl(2, 3, 5)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1) + E^(w1 + w3)
Bundle
U_3^v + U_2^v ot Sigma_(2,1) U_3^v
ZL62 label
45.0T
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 147
$\chi(X, \mathrm{T}_X)$ = -13

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 147
$\mathrm{c}_1^2\mathrm{c}_2$ = 126
$\mathrm{c}_2^2$ = 116
$\mathrm{c}_1\mathrm{c}_3$ = 34
$\mathrm{c}_4$ = 19
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 13 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 + 13 0 x0 + 13 x1
$\mathrm{HH}^{ 3 }$ = x1 + 31 0 0 x1 + 8 23
$\mathrm{HH}^{ 4 }$ = 36 0 0 0 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 83t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(426)
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( t + 1 \right) ^{2} \left( 4t^{2} + 1 \right) \left( 484t^{5} - 1408t^{4} + 1141t^{3} + 144t^{2} - t - 1 \right) \left( 2988051470976t^{17} + 13270592092736t^{16} + 14130998025408t^{15} - 5350849956144t^{14} - 16309283188440t^{13} - 17005500896472t^{12} - 4827950300048t^{11} + 781473741256t^{10} - 797285906112t^{9} - 4658879057714t^{8} - 2513738018895t^{7} - 1124710725483t^{6} + 9062255510t^{5} + 13272320576t^{4} - 1131456171t^{3} - 210463951t^{2} - 2020450t - 23226 \right)D^{6} + \left( t + 1 \right) \left( 98342750012762112t^{25} + 295744225578971136t^{24} - 218036726039373824t^{23} - 875924858989469696t^{22} + 948912994244864t^{21} + 502624094788140160t^{20} + 693158429587514880t^{19} - 245919564257064896t^{18} - 614627393933177784t^{17} - 310410519304439720t^{16} + 320815274898087624t^{15} + 110446579997354496t^{14} - 102212035291571280t^{13} - 152471168775653180t^{12} - 85536204849373529t^{11} - 16878458885591065t^{10} - 5243141318172661t^{9} + 37259486646332t^{8} + 60774145862018t^{7} - 17956391939701t^{6} - 2690700515439t^{5} - 58836555872t^{4} - 6521158581t^{3} - 849010684t^{2} - 5875542t - 46452 \right)D^{5} + \left( 665259779498096640t^{26} + 2913408423391600640t^{25} + 1634659924491990528t^{24} - 5779574851373852672t^{23} - 6654653347353786368t^{22} - 1307486628160113984t^{21} + 5415370494990306400t^{20} + 5594835372313507760t^{19} - 340959989206923912t^{18} - 3040588338201604360t^{17} + 868330388315137256t^{16} + 2544078814612343552t^{15} + 249275476199494816t^{14} - 1044116752009342998t^{13} - 1087584337682656131t^{12} - 349213750177958315t^{11} - 23683035622369872t^{10} + 3393904082774525t^{9} + 1238080335944835t^{8} + 167858144669321t^{7} - 20285910930135t^{6} - 7758355084849t^{5} - 458324382732t^{4} - 1494821140t^{3} + 770060449t^{2} + 6077520t + 23226 \right)D^{4} + t \left( 2285022720884766720t^{25} + 10669944316370042880t^{24} + 8433743888512250880t^{23} - 15681124517603421184t^{22} - 23272642818575242752t^{21} - 13972160853351137536t^{20} + 10758615127470144000t^{19} + 18653643305984976256t^{18} + 2625394538678456472t^{17} - 10752849833784392032t^{16} + 698321539066038240t^{15} + 6068372933980472200t^{14} + 1358799406473740672t^{13} - 1931021114542879300t^{12} - 2567061293995941787t^{11} - 689056524855919390t^{10} + 20103705539905454t^{9} + 25633887754137131t^{8} + 181085927106398t^{7} + 6149028554335t^{6} - 51004327560688t^{5} - 10907379546963t^{4} - 276604011383t^{3} - 5940424387t^{2} - 61729174t - 15958 \right)D^{3} + 2 t ^{2} \left( 2094122088507052032t^{24} + 10250990755098936832t^{23} + 9687337364040064256t^{22} - 11528753859187371136t^{21} - 21457193006282277376t^{20} - 19290331634695661760t^{19} + 3371078915153920432t^{18} + 14588388025255695904t^{17} + 2095016017141065920t^{16} - 11987121492191943544t^{15} - 763246162944834748t^{14} + 4317008108404715064t^{13} + 2078500890635195884t^{12} - 715792948467121560t^{11} - 1704487890365172753t^{10} - 381926766476032088t^{9} + 37472547947477211t^{8} + 17569705934379583t^{7} - 930099641925544t^{6} - 34193861095381t^{5} - 12826426033477t^{4} - 4576204421563t^{3} - 111458583903t^{2} - 3035261623t - 23813706 \right)D^{2} + 8 t ^{2} \left( 483036448592096256t^{24} + 2448504606599825664t^{23} + 2580646432949174656t^{22} - 2162438895203077568t^{21} - 4978918608183410880t^{20} - 5662679198242450560t^{19} - 563463637001205512t^{18} + 2760275108062646152t^{17} + 236511818304339412t^{16} - 3292863552811251392t^{15} - 388590086110814278t^{14} + 855196373196049288t^{13} + 668842542028745264t^{12} - 1528877456879444t^{11} - 309437875429122735t^{10} - 56241458396096870t^{9} + 9141924735691685t^{8} + 2474302328402267t^{7} - 243884342597822t^{6} - 4138390187273t^{5} - 357648116068t^{4} - 534403223882t^{3} - 14562173473t^{2} - 397183328t - 2345826 \right)D + 64 t ^{2} \left( 21693253679285760t^{24} + 112753833054192000t^{23} + 127438682644920768t^{22} - 80567802519596128t^{21} - 224828095209715008t^{20} - 296997545930740688t^{19} - 75435577380970620t^{18} + 100924624367573996t^{17} + 509085203726354t^{16} - 167783543240057544t^{15} - 24415203966747045t^{14} + 35134650898907390t^{13} + 36938294709858832t^{12} + 4929291495668608t^{11} - 11842516275552626t^{10} - 1756033526762444t^{9} + 403937745936491t^{8} + 61986403952137t^{7} - 9345888758503t^{6} + 25722993063t^{5} + 14493591486t^{4} - 13562349552t^{3} - 450038522t^{2} - 10816048t - 46452 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5)

|λ| = 14.24974 λ ≈ 14.24974; multiplicity 1; Jordan blocks 1 λ ≈ -8.186205 - 4.929476i; multiplicity 1; Jordan blocks 1 λ ≈ -8.186205 + 4.929476i; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ 0.5613353 - 0.2384702i; multiplicity 1; Jordan blocks 1 λ ≈ 0.5613353 + 0.2384702i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1 2
Sector dimension
10
Spectral radius
$14.24974$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14.24974$11
$-8.186205 - 4.929476\mathrm{i}$11
$-8.186205 + 4.929476\mathrm{i}$11
$0 - 2\mathrm{i}$11
$0 + 2\mathrm{i}$11
$-1$22
$0.5613353 - 0.2384702\mathrm{i}$11
$0.5613353 + 0.2384702\mathrm{i}$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda^{2} + 4\right) \left(\lambda^{5} + \lambda^{4} - 144 \lambda^{3} - 1141 \lambda^{2} + 1408 \lambda - 484\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda^{2} + 4\right) \left(\lambda^{5} + \lambda^{4} - 144 \lambda^{3} - 1141 \lambda^{2} + 1408 \lambda - 484\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 15 & 39 & 9 & 33 & 40 & 44 & 110 \\ 1 & 0 & 3 & 4 & -4 & 2 & -6 & 18 & 9 & -8 \\ 1 & 0 & -1 & 8 & 16 & 4 & 12 & 9 & 9 & 32 \\ 0 & 1 & 0 & 1 & 3 & 0 & 3 & 2 & 4 & 12 \\ 0 & 1 & 2 & -1 & -1 & 0 & 0 & 6 & 6 & 6 \\ 0 & 2 & 1 & -2 & 2 & 0 & 2 & -4 & -2 & 6 \\ 0 & 0 & 1 & 2 & 0 & 0 & -1 & 4 & 2 & 0 \\ 0 & 0 & 0 & 1 & 7 & \frac{5}{3} & \frac{20}{3} & -1 & 0 & \frac{32}{3} \\ 0 & 0 & 0 & 2 & -2 & \frac{2}{3} & -\frac{10}{3} & 2 & 0 & -\frac{16}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$