Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-272-60-b

·

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

Defining data
Ambient
ℙ3P3 × ℙ6P6
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_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
2 copies of E^(w1 + w4) + E^(w3 + w4)
Bundle
2 copies of O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
30600.wFw2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 60
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 272
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 272
$\mathrm{c}_1^2\mathrm{c}_2$ = 164
$\mathrm{c}_2^2$ = 103
$\mathrm{c}_1\mathrm{c}_3$ = 42
$\mathrm{c}_4$ = 15
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 9 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 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 50 0 0 x1 - 22 72
$\mathrm{HH}^{ 4 }$ = 60 0 0 0 0 60
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 55t + 160t^{2} + 55t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(113)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(160)

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( 229t^{8} - 1976t^{7} + 4533t^{6} - 6844t^{5} - 1306t^{4} - 544t^{3} - 13t^{2} + 3t + 1 \right) \left( 21238797237714t^{13} + 69609894358032t^{12} + 144149746280106t^{11} + 22091916144571t^{10} - 45191460811833t^{9} + 111498885642382t^{8} + 29167342363866t^{7} + 5370208647346t^{6} - 6676520553728t^{5} + 828056042491t^{4} - 65980435446t^{3} + 4509427030t^{2} + 4104057t + 687888 \right)D^{6} + \left( 82682637646420602t^{21} - 258650238898589328t^{20} - 432013360548837984t^{19} - 1742792634025189180t^{18} + 3385756471783383313t^{17} - 3771570563701049816t^{16} - 8171942569541103521t^{15} + 8761749449307561158t^{14} - 5596140921426592420t^{13} - 2377203482415214572t^{12} - 1384074306621744223t^{11} + 564489230738279691t^{10} - 19577992393411920t^{9} + 12975403142126810t^{8} - 2488498765846877t^{7} + 175770417959078t^{6} + 24561097190742t^{5} - 5037555961432t^{4} + 325893302805t^{3} - 18079051213t^{2} - 8184843t - 1375776 \right)D^{5} + \left( 559323725255198190t^{21} - 797109655415660244t^{20} - 2844722357498752542t^{19} - 13200728173318841603t^{18} + 18410320932429240437t^{17} + 3060016500005859779t^{16} - 42438352129596782219t^{15} + 38749481173018264300t^{14} - 18384971587735978160t^{13} - 6315091449864757896t^{12} - 6280875485767876994t^{11} + 3309929666639083503t^{10} - 220763632401701810t^{9} + 31064794991856134t^{8} - 4533325440711599t^{7} + 466336945203857t^{6} - 147129653573817t^{5} + 6883126170991t^{4} - 385914416155t^{3} + 17753880271t^{2} + 66283227t + 687888 \right)D^{4} + t \left( 1921155404137419870t^{20} - 268350162555033720t^{19} - 6623048798879223396t^{18} - 47072122555502195200t^{17} + 39290986987179439967t^{16} + 44732706562153744148t^{15} - 123030647513308560655t^{14} + 81072176737423960874t^{13} - 24855255906808485108t^{12} - 3898852728361158486t^{11} - 17657680213359823777t^{10} + 8831414262841660831t^{9} - 925978713105862508t^{8} + 40917146487922766t^{7} - 6265577189148771t^{6} + 94789938777696t^{5} - 115448186300728t^{4} - 1003371600982t^{3} - 37978350229t^{2} - 405760719t - 63225 \right)D^{3} + 2 t ^{2} \left( 1760653813412015172t^{19} + 1436652683421029298t^{18} - 2507077735648178595t^{17} - 41667160613640816688t^{16} + 18224762879092114149t^{15} + 50458787920136852298t^{14} - 97252381873489017980t^{13} + 44130817947841871598t^{12} - 4339645109853552926t^{11} + 4462828085546668605t^{10} - 14102024623824662402t^{9} + 6054833658125560436t^{8} - 867092880451785642t^{7} + 38030628446713418t^{6} - 2362527599585725t^{5} - 310669168964521t^{4} - 57779701868371t^{3} - 421230106290t^{2} - 23901921268t - 156743808 \right)D^{2} + 4 t ^{2} \left( 812235322761896502t^{19} + 1230041592101891142t^{18} + 379965082471927407t^{17} - 17846775235184650846t^{16} + 2784402814545540678t^{15} + 24146864872843261878t^{14} - 38993875231586465946t^{13} + 11943728831589294252t^{12} + 2036282762325234418t^{11} + 3761798267282731911t^{10} - 5780566982682736178t^{9} + 2081908313404824665t^{8} - 376883719424131448t^{7} + 21241042354358511t^{6} - 775927584762581t^{5} - 199052315294160t^{4} - 14748376078780t^{3} - 139426613316t^{2} - 7544442404t - 32101440 \right)D + 48 t ^{2} \left( 24318422837182530t^{19} + 48812446938776598t^{18} + 48582070640198487t^{17} - 490778525055066136t^{16} - 23766089923954077t^{15} + 717563165468888718t^{14} - 1029715131785955006t^{13} + 207975472628811906t^{12} + 112544987074434385t^{11} + 134658888990824157t^{10} - 156134672259837008t^{9} + 47447283729901244t^{8} - 10349064318843371t^{7} + 722083608393228t^{6} - 24735082704638t^{5} - 6110482240950t^{4} - 254639595715t^{3} - 3700928589t^{2} - 161139731t - 458592 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ6P6

|λ| = 8.718246 λ ≈ 8.718246; multiplicity 1; Jordan blocks 1 λ ≈ -4.980171 - 5.712035i; multiplicity 1; Jordan blocks 1 λ ≈ -4.980171 + 5.712035i; multiplicity 1; Jordan blocks 1 λ ≈ -1.19797 - 3.74917i; multiplicity 1; Jordan blocks 1 λ ≈ -1.19797 + 3.74917i; multiplicity 1; Jordan blocks 1 λ ≈ 0.2390469 - 0.3570012i; multiplicity 1; Jordan blocks 1 λ ≈ 0.2390469 + 0.3570012i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1599429; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$8.71824572$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.718246$11
$-4.980171 - 5.712035\mathrm{i}$11
$-4.980171 + 5.712035\mathrm{i}$11
$-1.19797 - 3.74917\mathrm{i}$11
$-1.19797 + 3.74917\mathrm{i}$11
$0.2390469 - 0.3570012\mathrm{i}$11
$0.2390469 + 0.3570012\mathrm{i}$11
$0.1599429$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{8} + 3 \lambda^{7} - 13 \lambda^{6} - 544 \lambda^{5} - 1306 \lambda^{4} - 6844 \lambda^{3} + 4533 \lambda^{2} - 1976 \lambda + 229\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{8} + 3 \lambda^{7} - 13 \lambda^{6} - 544 \lambda^{5} - 1306 \lambda^{4} - 6844 \lambda^{3} + 4533 \lambda^{2} - 1976 \lambda + 229\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 12 & 9 & 12 & 0 & 0 & 20 \\ 1 & -1 & 0 & 2 & 0 & 6 & 3 & 6 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 & 21 & 12 & 0 \\ 0 & 2 & 1 & -1 & -2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 6 & 2 & 0 \\ 0 & 0 & 2 & 1 & 2 & 0 & 0 & 0 & 12 \\ 0 & 0 & 0 & 2 & -\frac{1}{5} & \frac{21}{5} & \frac{3}{5} & \frac{6}{5} & -\frac{8}{5} \\ 0 & 0 & 0 & 1 & \frac{13}{5} & -\frac{8}{5} & -\frac{4}{5} & -\frac{8}{5} & \frac{24}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{11}{4} & \frac{7}{4} & 0\end{pmatrix}\notag\end{equation}$$