Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-56-18-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w3 + w4)
Bundle
O_1(1) box U_2^v(1)
ZL62 label
2031.x4300
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 56
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 56
$\mathrm{c}_1^2\mathrm{c}_2$ = 92
$\mathrm{c}_2^2$ = 157
$\mathrm{c}_1\mathrm{c}_3$ = -2
$\mathrm{c}_4$ = 61
Hodge diamond
1
00
0 2 0
0 0 0 0
0 4 47 4 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 36 0 36 0
$\mathrm{HH}^{ 3 }$ = 4 0 0 4 0
$\mathrm{HH}^{ 4 }$ = 28 0 0 10 0 18
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 28t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(679)
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( 4t + 1 \right) ^{5} \left( 53062t^{5} + 35675t^{4} + 8029t^{3} + 601t^{2} + 19t - 1 \right) \left( 80439069224960t^{11} + 5566720538624t^{10} - 104981361482880t^{9} - 95739945814144t^{8} - 40665895138080t^{7} - 10049029672320t^{6} - 1552338008392t^{5} - 154401163256t^{4} - 9869354054t^{3} - 349564684t^{2} - 1672713t - 4508 \right)D^{6} + \left( 4t + 1 \right) ^{4} \left( 358533662862045511680t^{17} + 251008499890735325184t^{16} - 453633257001371943936t^{15} - 818073506284160255232t^{14} - 609201808213386491520t^{13} - 274388110944337338176t^{12} - 83366002745649442016t^{11} - 17938805995188879472t^{10} - 2802928995425213384t^{9} - 321761639883243900t^{8} - 27102775165076718t^{7} - 1637494192411916t^{6} - 67216913449861t^{5} - 1798883798631t^{4} - 42876270255t^{3} - 1378178487t^{2} - 4973059t - 9016 \right)D^{5} + \left( 4t + 1 \right) ^{3} \left( 11951122095401517056000t^{18} + 8568365029815346462720t^{17} - 16278693777526801330176t^{16} - 30423510136074078429184t^{15} - 23691558126691286318336t^{14} - 11249494697858540804224t^{13} - 3637310533924309389504t^{12} - 842246349872562980640t^{11} - 143469132127585173680t^{10} - 18224147737041463304t^{9} - 1726989233435695060t^{8} - 119351905747382150t^{7} - 5614830375409292t^{6} - 151859198587021t^{5} - 1290884301431t^{4} + 30197582156t^{3} + 1241150148t^{2} + 4097135t + 4508 \right)D^{4} + t \left( 4t + 1 \right) ^{2} \left( 200778851202745486540800t^{18} + 157363220828398216151040t^{17} - 282279438720609434746880t^{16} - 571289344869836234649600t^{15} - 475324526936877982587904t^{14} - 242196213514802304050944t^{13} - 84689847603022352612736t^{12} - 21414911408178852650176t^{11} - 4028479962820699204768t^{10} - 572525554875731011664t^{9} - 61739922764697109624t^{8} - 4993845179065910324t^{7} - 291377339656769578t^{6} - 11272426196214548t^{5} - 247593311449087t^{4} - 2484621360087t^{3} - 17396378923t^{2} - 74131246t - 25921 \right)D^{3} + \left( 4t + 1 \right) t ^{2} \left( 1774502608725217252474880t^{18} + 1585713762207890042519552t^{17} - 2475903484706578340773888t^{16} - 5637091149231475943407616t^{15} - 5089571129271710159142912t^{14} - 2815318149079593763729408t^{13} - 1074876851779494319773696t^{12} - 299125098963067154138880t^{11} - 62508879312843392405632t^{10} - 9975574455565399155584t^{9} - 1224406996471404380032t^{8} - 115042816837675026992t^{7} - 8082681117589859624t^{6} - 402552929358875152t^{5} - 12878104893407780t^{4} - 226320336959710t^{3} - 2063044926388t^{2} - 20001957619t - 63746984 \right)D^{2} + 4 t ^{2} \left( 1927476971546356670791680t^{19} + 2006316003339797878800384t^{18} - 2549961680333275448737792t^{17} - 6798582446603715298525184t^{16} - 6740471628422798644051968t^{15} - 4079348179353915697709056t^{14} - 1710522312002734841363456t^{13} - 526131950804091301973248t^{12} - 122472577951273188041216t^{11} - 21969448391931983455872t^{10} - 3064971732971674075104t^{9} - 332587302516891286832t^{8} - 27713035406551763024t^{7} - 1715696178024284176t^{6} - 74071627134995066t^{5} - 2003257653016326t^{4} - 29246675377367t^{3} - 252650297564t^{2} - 2286714850t - 6112848 \right)D + 32 t ^{2} \left( 98340661813589626060800t^{19} + 95539309173517335920640t^{18} - 139718364727062721855488t^{17} - 346290434532582056951808t^{16} - 332666074590684706947072t^{15} - 195982023523048618696704t^{14} - 80040023839792371992064t^{13} - 23966523288776718970880t^{12} - 5426329689792639639296t^{11} - 945786746854056145408t^{10} - 128020642284311656576t^{9} - 13441995522536783200t^{8} - 1077720517847370240t^{7} - 63503146121185424t^{6} - 2560305503419434t^{5} - 62783071699278t^{4} - 810199414829t^{3} - 6844517092t^{2} - 47434303t - 99176 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 4)

|λ| = 39.77392 λ ≈ 39.77392; multiplicity 1; Jordan blocks 1 λ ≈ -7.256662 - 8.422941i; multiplicity 1; Jordan blocks 1 λ ≈ -7.256662 + 8.422941i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 5; Jordan blocks 1, 1, 3 λ ≈ -3.130299 - 0.9971949i; multiplicity 1; Jordan blocks 1 λ ≈ -3.130299 + 0.9971949i; multiplicity 1; Jordan blocks 1 5
Sector dimension
10
Spectral radius
$39.7739215$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$39.77392$11
$-7.256662 - 8.422941\mathrm{i}$11
$-7.256662 + 8.422941\mathrm{i}$11
$-4$51, 1, 3
$-3.130299 - 0.9971949\mathrm{i}$11
$-3.130299 + 0.9971949\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{5} \left(\lambda^{5} - 19 \lambda^{4} - 601 \lambda^{3} - 8029 \lambda^{2} - 35675 \lambda - 53062\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda^{5} - 19 \lambda^{4} - 601 \lambda^{3} - 8029 \lambda^{2} - 35675 \lambda - 53062\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 34 & 54 & 714 & 282 & 480 & 396 & 5448 & 4584 & 22620 \\ 1 & -1 & 8 & 60 & 26 & 48 & 38 & 594 & 507 & 2592 \\ 1 & 3 & 2 & 88 & 42 & 52 & 42 & 702 & 591 & 2856 \\ 0 & 1 & 1 & 1 & 0 & 6 & 5 & 42 & 36 & 234 \\ 0 & 1 & 0 & 8 & -1 & 4 & 3 & 42 & 34 & 180 \\ 0 & 0 & 1 & 3 & 3 & 0 & 2 & 48 & 42 & 198 \\ 0 & 0 & 1 & 3 & 2 & 0 & -3 & 30 & 26 & 126 \\ 0 & 0 & 0 & -\frac{7}{3} & -\frac{10}{3} & \frac{10}{3} & \frac{7}{3} & -14 & -\frac{23}{3} & -6 \\ 0 & 0 & 0 & 6 & 5 & -2 & -1 & 24 & 15 & 60 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$