Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-322-69-a

·

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

Defining data
Ambient
Fl(1, 2, 6) × ℙ4P4
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_6 (marked)node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w1) + E^(w3) + E^(w5 + w6)
Bundle
O_1(1,0) + Q_1,1^v ot U_1,2^v ot U_1,3^v + wedge^3 Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1)
ZL62 label
4052.64w7
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,4;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 69
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 322
$\chi(X, \mathrm{T}_X)$ = 8

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 322
$\mathrm{c}_1^2\mathrm{c}_2$ = 172
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 8 0 x0 - 8 x1
$\mathrm{HH}^{ 3 }$ = x1 + 44 0 0 x1 - 40 84
$\mathrm{HH}^{ 4 }$ = 69 0 0 0 0 69
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 64t + 192t^{2} + 64t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(18)
Coates et al. (2020) Obro(4,32)
Coates–Kasprzyk–Prince (2015) CKP(25)

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) \left( 17t^{2} + 2t + 1 \right) \left( 11t^{3} + 18t^{2} - t + 1 \right) \left( 176t^{4} + 288t^{3} + 32t^{2} - 2t - 1 \right) \left( 30204320314080t^{17} + 854407991331904t^{16} + 3326457732193488t^{15} + 6262704961672960t^{14} + 6687161269031338t^{13} + 4287542223784788t^{12} + 1902956450746962t^{11} + 927758009882065t^{10} + 508153538481270t^{9} + 170158223541909t^{8} + 25546795301128t^{7} + 1442573891170t^{6} + 81976582034t^{5} + 31331901974t^{4} + 1259858350t^{3} + 77122349t^{2} + 1195830t + 36225 \right)D^{6} + 2 \left( 10437888196858510080t^{27} + 346420463396749950208t^{26} + 2412815616346508883840t^{25} + 8357107992121261849312t^{24} + 17781283519700094322080t^{23} + 25340077477976832758288t^{22} + 25318074692820287424760t^{21} + 18553117104281005437682t^{20} + 10858632155808797927307t^{19} + 5795244218853140267875t^{18} + 2961387793705792670165t^{17} + 1288080246445993636244t^{16} + 423662017107926134785t^{15} + 104080002348157261007t^{14} + 19941097639291389410t^{13} + 2959510609671413309t^{12} + 357961500730925139t^{11} + 47634789132155469t^{10} + 8983977260665226t^{9} + 1495819264787535t^{8} + 163660156065719t^{7} + 7800914322149t^{6} + 483186249226t^{5} + 97123055069t^{4} + 3171737050t^{3} + 151200988t^{2} + 1648845t + 36225 \right)D^{5} + \left( 173964803280975168000t^{27} + 5844758802867848568320t^{26} + 38718785582639539190528t^{25} + 127871302774736282458432t^{24} + 260112111794532725372224t^{23} + 354457390473677851865056t^{22} + 338788203130455062884752t^{21} + 239268884651243032578508t^{20} + 138646173658552109413498t^{19} + 76093604162462122856270t^{18} + 39849225226252330977816t^{17} + 16980559208370661634009t^{16} + 5175434849289162467811t^{15} + 1098475763787735117019t^{14} + 172163929071457851390t^{13} + 21041600237295609830t^{12} + 2074320670512019567t^{11} + 191728343609422595t^{10} + 16552253324575690t^{9} - 1465899994529776t^{8} - 253358387324775t^{7} + 228485300153t^{6} + 1569438458618t^{5} - 131526697530t^{4} - 4799195845t^{3} - 285371237t^{2} - 3800010t - 36225 \right)D^{4} + 2 t \left( 365326086890047852800t^{26} + 12391282767343808500480t^{25} + 78673881020044158823040t^{24} + 249556108547605048782240t^{23} + 488019545866244796234592t^{22} + 638776769240086029399408t^{21} + 586845593558651987728456t^{20} + 402613900084989278726278t^{19} + 234142902068063686958841t^{18} + 133190365283038596048517t^{17} + 70912722180797127263511t^{16} + 29395429332009459274392t^{15} + 8361532251701923504341t^{14} + 1566731831308409941358t^{13} + 205567634368486251141t^{12} + 23131387637148171125t^{11} + 2619336887795423518t^{10} + 292571160551349845t^{9} + 41668988456304994t^{8} + 2988757149285390t^{7} + 186206145618841t^{6} + 10179031940632t^{5} + 536652063185t^{4} + 14427226167t^{3} + 397619827t^{2} + 2722200t + 3825 \right)D^{3} + 4 t ^{2} \left( 403598343611862389760t^{25} + 13788554521229995367168t^{24} + 84583359591399353963520t^{23} + 259594447224376604386496t^{22} + 491041035289219660466016t^{21} + 620764388988455909753648t^{20} + 551148075436572061900888t^{19} + 369760939405849649826172t^{18} + 217326579795809445433642t^{17} + 127770035729810131024508t^{16} + 68470163100260610504947t^{15} + 27534236546146345850210t^{14} + 7376598225713942927971t^{13} + 1247538867569364789140t^{12} + 139498454756295155334t^{11} + 15292094361877018212t^{10} + 2110172684810561361t^{9} + 226097847460079973t^{8} + 27369681585146555t^{7} + 1590110139019710t^{6} + 94152994494893t^{5} + 6609627393830t^{4} + 340685456528t^{3} + 9554354012t^{2} + 237552057t + 828345 \right)D^{2} + 8 t ^{2} \left( 219195652134028711680t^{25} + 7528067952366431185408t^{24} + 44980233110407074451840t^{23} + 134561131543064752166432t^{22} + 247869724716976921554752t^{21} + 304593634114044952766304t^{20} + 262969872795464641201152t^{19} + 173509746858655649351030t^{18} + 103292478599138920197317t^{17} + 62328941662347267359327t^{16} + 33384175889368176727374t^{15} + 13043602214892537064173t^{14} + 3326057396894893759218t^{13} + 519674794492332524146t^{12} + 51154064294579664551t^{11} + 5775450686110347722t^{10} + 944583027533807439t^{9} + 91048270448630171t^{8} + 8987062901404216t^{7} + 432446001799677t^{6} + 29598467542244t^{5} + 2447326169556t^{4} + 117295462907t^{3} + 3360473726t^{2} + 71389550t + 96600 \right)D + 48 t ^{3} \left( 14911268852655014400t^{24} + 513926844028039946240t^{23} + 3015073573594738244864t^{22} + 8858125147867035941056t^{21} + 16006727812946589608576t^{20} + 19261174062581447102080t^{19} + 16283984376708622141568t^{18} + 10619567953236074488292t^{17} + 6395127990371776243674t^{16} + 3930666192061264664474t^{15} + 2098143214031052424978t^{14} + 800564409305073411246t^{13} + 196526504877884528359t^{12} + 28969026690487342945t^{11} + 2598574238134156891t^{10} + 310981308966577644t^{9} + 56758978024887833t^{8} + 4853835872273256t^{7} + 383944505572388t^{6} + 15565331639506t^{5} + 1450362039457t^{4} + 126306692137t^{3} + 5684787555t^{2} + 162643140t + 2934225 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6) × ℙ4P4

|λ| = 7.59921 λ ≈ 7.59921; multiplicity 1; Jordan blocks 1 λ ≈ -4.469104 - 3.881411i; multiplicity 1; Jordan blocks 1 λ ≈ -4.469104 + 3.881411i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7907061 - 4.277176i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7907061 + 4.277176i; multiplicity 1; Jordan blocks 1 λ ≈ -1 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.6610007; multiplicity 1; Jordan blocks 1 λ ≈ -0.5814122; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
13
Spectral radius
$7.59920951$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.59921$11
$-4.469104 - 3.881411\mathrm{i}$11
$-4.469104 + 3.881411\mathrm{i}$11
$0.7907061 - 4.277176\mathrm{i}$11
$0.7907061 + 4.277176\mathrm{i}$11
$-1 - 4\mathrm{i}$21, 1
$-1 + 4\mathrm{i}$21, 1
$-1$21, 1
$-0.6610007$11
$-0.5814122$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{2} + 2 \lambda + 17\right)^{2} \left(\lambda^{3} - \lambda^{2} + 18 \lambda + 11\right) \left(\lambda^{4} + 2 \lambda^{3} - 32 \lambda^{2} - 288 \lambda - 176\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{2} + 2 \lambda + 17\right) \left(\lambda^{3} - \lambda^{2} + 18 \lambda + 11\right) \left(\lambda^{4} + 2 \lambda^{3} - 32 \lambda^{2} - 288 \lambda - 176\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 6 & 6 & 6 & 6 & 4 & 4 & 8 & 20 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 & 8 \\ 1 & 1 & 0 & 1 & 0 & 0 & -2 & -2 & -4 & 3 & 3 & 3 & -8 \\ 1 & 0 & 0 & -1 & 0 & 0 & 4 & 0 & 4 & 0 & 0 & 0 & 8 \\ 0 & 1 & 2 & -1 & 0 & 1 & 0 & 1 & -2 & -4 & -2 & -6 & -6 \\ 0 & 1 & 0 & 1 & 0 & -2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 1 & -1 & 0 & -1 & 0 & 2 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 & 1 & 0 & -1 & -1 & 2 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & 1 & 5 & 5 & 2 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 5 & 1 & 6 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -3 & -1 & 0 & 4 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{5}{2} & 3 & 0\end{pmatrix}\notag\end{equation}$$