Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-240-54-b

·

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

Defining data
Ambient
ℙ2P2 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(2w3) + E^(w1 + w3) + E^(w2 + w3)
Bundle
O_2(2) + O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
20600.Aw7w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 240
$\chi(X, \mathrm{T}_X)$ = -7

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 240
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 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 + 7 0 x0 + 7 x1
$\mathrm{HH}^{ 3 }$ = x1 + 46 0 0 x1 - 13 59
$\mathrm{HH}^{ 4 }$ = 54 0 0 0 0 54
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 49t + 140t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(222)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(300)

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( 4528t^{6} + 5680t^{5} + 168t^{4} + 568t^{3} + 47t^{2} - 2t - 1 \right) \left( 1526202893904t^{13} + 7614395240184t^{12} + 20199506451612t^{11} + 29375376614788t^{10} + 25205715852240t^{9} + 8696770587798t^{8} + 1599233188853t^{7} + 2397215023887t^{6} + 1295149932113t^{5} + 69665771055t^{4} - 9021962218t^{3} + 680571872t^{2} + 1902552t + 82080 \right)D^{6} + \left( 89838407146765056t^{20} + 622907055334492416t^{19} + 2193854116929222720t^{18} + 4664646650944258816t^{17} + 6373880858804583888t^{16} + 5462772576177940160t^{15} + 2779138234948728396t^{14} + 1059016992700049700t^{13} + 666831899320450492t^{12} + 416423033779406224t^{11} + 127488065135910357t^{10} + 18865868858718559t^{9} + 3954036716434010t^{8} + 349563573308100t^{7} + 4482829707789t^{6} + 11641336548587t^{5} + 524808054276t^{4} - 44176944890t^{3} + 2738741432t^{2} + 5215176t + 164160 \right)D^{5} + \left( 464740990816919232t^{20} + 3119260147133102496t^{19} + 10871635397747756208t^{18} + 22741340086743476720t^{17} + 30606341851735742208t^{16} + 25137149812755417600t^{15} + 12019833862444736528t^{14} + 4990282914560154612t^{13} + 3779403393976572156t^{12} + 2288518614215886862t^{11} + 608648467089867787t^{10} + 47978778590111516t^{9} + 3111605680708741t^{8} + 525041804433676t^{7} - 94996304799639t^{6} - 5500259515484t^{5} - 637465407013t^{4} + 50898809770t^{3} - 2730171856t^{2} - 12490536t - 82080 \right)D^{4} + t \left( 1226639789888522880t^{19} + 8040252418128292416t^{18} + 27770546384308100256t^{17} + 57359784880128051296t^{16} + 76499661347430757264t^{15} + 60590608610139095392t^{14} + 28076767696400555980t^{13} + 13566642081756494708t^{12} + 11618874667940852172t^{11} + 6581789253450553508t^{10} + 1531778862734909163t^{9} + 70629198306299713t^{8} - 6976251641977506t^{7} + 1352110370610206t^{6} - 14936526851675t^{5} + 21993717214199t^{4} + 287876234642t^{3} + 7123617826t^{2} + 95966944t + 12312 \right)D^{3} + 2 t ^{2} \left( 873332977167110304t^{18} + 5627175056600214576t^{17} + 19295272307170252008t^{16} + 39505259716090326888t^{15} + 52469823879252578936t^{14} + 40377404521944112492t^{13} + 18459174386194134722t^{12} + 10356466926330280294t^{11} + 9498120037810492082t^{10} + 5039233658242020658t^{9} + 1036943518309683680t^{8} + 28859575176108524t^{7} - 7906950456914091t^{6} + 1338436983818945t^{5} + 57903730605287t^{4} + 11323575872445t^{3} + 103005049900t^{2} + 4310676870t + 37269840 \right)D^{2} + 8 t ^{2} \left( 159376789601713008t^{18} + 1014210866750949960t^{17} + 3458024039276859132t^{16} + 7038890782897259916t^{15} + 9344209943575179484t^{14} + 7044489879120666618t^{13} + 3210694973501259487t^{12} + 2040618260054648801t^{11} + 1943706753732975905t^{10} + 975665971598294961t^{9} + 180178326876214708t^{8} + 3190547648445468t^{7} - 1317984858809996t^{6} + 320182404266140t^{5} + 17794941919468t^{4} + 1473687050948t^{3} + 14162373280t^{2} + 659469984t + 3830400 \right)D + 96 t ^{2} \left( 3887238770773488t^{18} + 24512277982795080t^{17} + 83211421427246748t^{16} + 168737893119395148t^{15} + 224398502235336772t^{14} + 166956968597424906t^{13} + 76288478054409667t^{12} + 53425959542992509t^{11} + 52003067674551671t^{10} + 24967412641844065t^{9} + 4201038384678623t^{8} + 48944570371409t^{7} - 22884185749977t^{6} + 9427708870915t^{5} + 536199981752t^{4} + 25886724496t^{3} + 340066496t^{2} + 13747944t + 54720 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ6P6

|λ| = 9.719613 λ ≈ 9.719613; multiplicity 1; Jordan blocks 1 λ ≈ -6.083416 - 5.179683i; multiplicity 1; Jordan blocks 1 λ ≈ -6.083416 + 5.179683i; multiplicity 1; Jordan blocks 1 λ ≈ 0.6095193 - 3.013907i; multiplicity 1; Jordan blocks 1 λ ≈ 0.6095193 + 3.013907i; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 1; Jordan blocks 1 λ ≈ -0.7718204; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 2 2
Sector dimension
9
Spectral radius
$9.71961286$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.719613$11
$-6.083416 - 5.179683\mathrm{i}$11
$-6.083416 + 5.179683\mathrm{i}$11
$0.6095193 - 3.013907\mathrm{i}$11
$0.6095193 + 3.013907\mathrm{i}$11
$-1$11
$-0.7718204$11
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right) \left(\lambda^{6} + 2 \lambda^{5} - 47 \lambda^{4} - 568 \lambda^{3} - 168 \lambda^{2} - 5680 \lambda - 4528\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right) \left(\lambda^{6} + 2 \lambda^{5} - 47 \lambda^{4} - 568 \lambda^{3} - 168 \lambda^{2} - 5680 \lambda - 4528\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 12 & 6 & 12 & 0 & 0 & 40 \\ 1 & -1 & 0 & 4 & 0 & 8 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 4 & 24 & 12 & 0 \\ 0 & 2 & 1 & -1 & -1 & 0 & 4 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 8 & 4 & 0 \\ 0 & 0 & 2 & 1 & 1 & 0 & 0 & 0 & 12 \\ 0 & 0 & 0 & 2 & 0 & 5 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 2 & -4 & -2 & -2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$