Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-70-21-d

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 5)
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_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
3 copies of E^(w4) + E^(2w1 + w4)
Bundle
3 copies of O_2(1) + O_1(2) box O_2(1)
ZL62 label
2041.333x327
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 70
$\chi(X, \mathrm{T}_X)$ = -30

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 70
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 143
$\mathrm{c}_1\mathrm{c}_3$ = 10
$\mathrm{c}_4$ = 49
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 37 3 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 + 30 0 x0 + 30 x1
$\mathrm{HH}^{ 3 }$ = x1 + 12 0 0 x1 + 12 0
$\mathrm{HH}^{ 4 }$ = 30 0 0 9 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 36t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(669)
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( 5t + 1 \right) ^{3} \left( 27851t^{6} + 24338t^{5} + 12944t^{4} + 2988t^{3} + 350t^{2} + 19t - 1 \right) \left( 534974213079025t^{13} + 1315662080395720t^{12} + 1269409792197891t^{11} + 668190272215134t^{10} + 219549079625307t^{9} + 48741372676140t^{8} + 7810013754743t^{7} + 960701525770t^{6} + 92973747528t^{5} + 6640987051t^{4} + 296499101t^{3} + 5936263t^{2} + 4857t + 154 \right)D^{6} + \left( 5t + 1 \right) ^{2} \left( 1564454514888712153875t^{20} + 5271488811220278464925t^{19} + 7829232690145058163565t^{18} + 7020160739547528267601t^{17} + 4293003954045415187649t^{16} + 1901644978786396135644t^{15} + 632036219580622803185t^{14} + 161628520718237511245t^{13} + 32524303409124787214t^{12} + 5269546762868522220t^{11} + 702264222007111839t^{10} + 77808879977420381t^{9} + 7081284832670255t^{8} + 509793921650620t^{7} + 27710418141754t^{6} + 1131178260675t^{5} + 39266713050t^{4} + 1268995650t^{3} + 24146990t^{2} + 12723t + 308 \right)D^{5} + \left( 5t + 1 \right) \left( 65185604787029673078125t^{21} + 224149610536437293197500t^{20} + 336552793562959931454450t^{19} + 302395623769955656942340t^{18} + 184554815200655135190226t^{17} + 81704778699196974641453t^{16} + 27302334956826547562263t^{15} + 7084735525431099847364t^{14} + 1463064992426118389345t^{13} + 246070147810290081485t^{12} + 34323323127593574371t^{11} + 3985056213638668360t^{10} + 377442220234501914t^{9} + 27849448546418052t^{8} + 1486941577732827t^{7} + 50177276447236t^{6} + 624042477587t^{5} - 26766178776t^{4} - 1434923967t^{3} - 23341964t^{2} - 27633t - 154 \right)D^{4} + t \left( 1368897700527623134640625t^{21} + 4840128881710783756040625t^{20} + 7450952640389910183901250t^{19} + 6838683929069527200391300t^{18} + 4257291955380449697587860t^{17} + 1926193690641863373047969t^{16} + 661272692102446956511066t^{15} + 177677701244814781465745t^{14} + 38342917521914155224495t^{13} + 6801548816636618402829t^{12} + 1009664087551867029093t^{11} + 126113691549152539471t^{10} + 13075108994823399342t^{9} + 1088660566322466581t^{8} + 69383370691690077t^{7} + 3183075963379829t^{6} + 96959838478058t^{5} + 1763155091919t^{4} + 18381524330t^{3} + 197927843t^{2} + 2505664t + 825 \right)D^{3} + 2 t ^{2} \left( 1512306031059088415412500t^{20} + 5228186835320311030596250t^{19} + 7774930169288167241487125t^{18} + 6832701128305626336759325t^{17} + 4046534143694346597916755t^{16} + 1734955897424546358071253t^{15} + 563919702355115300049311t^{14} + 143824556385287453975498t^{13} + 29639437239640442511204t^{12} + 5060776117070897601402t^{11} + 727278685061284725801t^{10} + 87812413739860412443t^{9} + 8705711087618647854t^{8} + 680695098298322317t^{7} + 39846101024682672t^{6} + 1638728116178993t^{5} + 43830226811464t^{4} + 723330157543t^{3} + 9585344946t^{2} + 151687314t + 889616 \right)D^{2} + 4 t ^{2} \left( 821338620316573880784375t^{20} + 2792083311115167049505625t^{19} + 4045036789202071822174125t^{18} + 3439440966507071867348175t^{17} + 1960747884313152682199685t^{16} + 806672675663356462242090t^{15} + 251435311693559941521087t^{14} + 61647534141623679818997t^{13} + 12279214634609983048692t^{12} + 2039298580701792661148t^{11} + 285963853403775196166t^{10} + 33568411644044822042t^{9} + 3200735888473336696t^{8} + 236937503172292205t^{7} + 12882694646840246t^{6} + 480980457156298t^{5} + 11386161235992t^{4} + 167681336288t^{3} + 2381642712t^{2} + 39260436t + 157696 \right)D + 144 t ^{2} \left( 9312229255289953296875t^{20} + 31293480955593720340625t^{19} + 44521530492139805337875t^{18} + 36991204940873483429725t^{17} + 20526121254905923233965t^{16} + 8198734127561760631718t^{15} + 2479490655476287000541t^{14} + 590935930443949578841t^{13} + 114889680087945925128t^{12} + 18708052196997840622t^{11} + 2575780784567534988t^{10} + 295666745430972610t^{9} + 27318332170752030t^{8} + 1935876615186209t^{7} + 99311893009986t^{6} + 3439633373296t^{5} + 74432241780t^{4} + 1030856872t^{3} + 15005784t^{2} + 211398t + 616 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 32.82396 λ ≈ 32.82396; multiplicity 1; Jordan blocks 1 λ ≈ -3.276726 - 5.462641i; multiplicity 1; Jordan blocks 1 λ ≈ -3.276726 + 5.462641i; multiplicity 1; Jordan blocks 1 λ ≈ -5.000497; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 3; Jordan blocks 3 λ ≈ -1.135006 - 1.701017i; multiplicity 1; Jordan blocks 1 λ ≈ -1.135006 + 1.701017i; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$32.8239622$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$32.82396$11
$-3.276726 - 5.462641\mathrm{i}$11
$-3.276726 + 5.462641\mathrm{i}$11
$-5.000497$11
$-5$33
$-1.135006 - 1.701017\mathrm{i}$11
$-1.135006 + 1.701017\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{6} - 19 \lambda^{5} - 350 \lambda^{4} - 2988 \lambda^{3} - 12944 \lambda^{2} - 24338 \lambda - 27851\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{6} - 19 \lambda^{5} - 350 \lambda^{4} - 2988 \lambda^{3} - 12944 \lambda^{2} - 24338 \lambda - 27851\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 28 & 44 & 426 & 123 & 198 & 2112 & 1820 & 10520 \\ 1 & 0 & 6 & 44 & 12 & 24 & 246 & 210 & 1312 \\ 1 & 3 & 2 & 56 & 18 & 24 & 288 & 255 & 1456 \\ 0 & 1 & 1 & 2 & 1 & 4 & 24 & 20 & 144 \\ 0 & 1 & 0 & 6 & -2 & 0 & 24 & 20 & 108 \\ 0 & 0 & \frac{5}{2} & \frac{15}{2} & \frac{5}{2} & 0 & 48 & 45 & 246 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & 3 & 6 & \frac{15}{2} & 56 \\ 0 & 0 & 0 & 1 & 1 & -\frac{8}{5} & -\frac{4}{5} & -4 & -\frac{48}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{5}{4} & 0\end{pmatrix}\notag\end{equation}$$