Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-272-60-d

·

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

Defining data
Ambient
ℙ2P2 × ℙ7P7
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_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(2w3) + 2 copies of E^(w2 + w3)
Bundle
O_2(2) + 2 copies of Q_1 box O_2(1)
ZL62 label
20700.Bw8w8
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;8) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 60
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 272
$\chi(X, \mathrm{T}_X)$ = -5

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 272
$\mathrm{c}_1^2\mathrm{c}_2$ = 164
$\mathrm{c}_2^2$ = 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 + 5 0 x0 + 5 x1
$\mathrm{HH}^{ 3 }$ = x1 + 50 0 0 x1 - 23 73
$\mathrm{HH}^{ 4 }$ = 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(167)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(233)

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( 44t^{3} - 48t^{2} - 12t - 1 \right) \left( 172t^{3} - 48t^{2} + 12t - 1 \right) \left( 343336313232t^{12} - 1378417144896t^{11} + 2644769069220t^{10} - 1923347843540t^{9} - 339222555264t^{8} + 1765587319307t^{7} - 1172261606364t^{6} + 255360546744t^{5} - 1607296368t^{4} - 10680119158t^{3} + 344155098t^{2} - 15129956t - 48960 \right)D^{6} + 2 \left( 16889399920508544t^{19} - 100734387181591680t^{18} + 283292024883579360t^{17} - 423567432422496032t^{16} + 289513853956939968t^{15} + 70402213966441632t^{14} - 319193468580753688t^{13} + 275852801710401720t^{12} - 113644992645454458t^{11} + 20801309762926018t^{10} + 242027484916152t^{9} - 778661921582577t^{8} + 95082623472365t^{7} - 14408497862214t^{6} + 798263512032t^{5} + 19979566654t^{4} - 33701924559t^{3} + 854512545t^{2} - 30284392t - 73440 \right)D^{5} + \left( 174740329946799936t^{19} - 989818178114126592t^{18} + 2688133469456420304t^{17} - 3725197616255776112t^{16} + 2045005737831632384t^{15} + 1539072633278921092t^{14} - 3647165084775830776t^{13} + 2807428745402597184t^{12} - 1056119977943910810t^{11} + 159699222169842100t^{10} + 17927734089143612t^{9} - 9579048700015037t^{8} + 667466167980173t^{7} - 29079957318252t^{6} - 1868058032640t^{5} - 16580251118t^{4} + 87845484338t^{3} - 1661076194t^{2} + 62895428t + 97920 \right)D^{4} + 2 t \left( 230605268145405120t^{18} - 1257620605271673984t^{17} + 3318685062159595392t^{16} - 4280481694791616480t^{15} + 1802377875734669888t^{14} + 2815394285176597404t^{13} - 5133418297480981836t^{12} + 3619458539265135288t^{11} - 1246329149494778919t^{10} + 146679631654263038t^{9} + 40154800095065264t^{8} - 14387597049281472t^{7} + 943324816146498t^{6} - 53418707319450t^{5} + 2798383506804t^{4} - 114963971032t^{3} - 2227267084t^{2} - 13023009t + 180 \right)D^{3} + 4 t ^{2} \left( 164184454996482096t^{17} - 871152560504024832t^{16} + 2247065190429510660t^{15} - 2722542478960051020t^{14} + 836954632517713968t^{13} + 2412549874536986611t^{12} - 3825968566814319388t^{11} + 2519916196542499668t^{10} - 793980913409234497t^{9} + 62567591971528047t^{8} + 38767224935527544t^{7} - 10959545840130272t^{6} + 676026698782978t^{5} - 31393674379764t^{4} + 994151468332t^{3} - 74091109416t^{2} - 578819600t - 3568640 \right)D^{2} + 8 t ^{2} \left( 59924890102573584t^{17} - 311756150737538496t^{16} + 790064065608569532t^{15} - 908702444191746468t^{14} + 191634897041441968t^{13} + 990527812131113793t^{12} - 1446289523119098080t^{11} + 903595287914052960t^{10} - 260969421394469168t^{9} + 9016103675439990t^{8} + 17440418381834276t^{7} - 4135142186638284t^{6} + 247382917108436t^{5} - 10861797140439t^{4} + 39293076512t^{3} - 21415229460t^{2} - 109478736t - 391680 \right)D + 96 t ^{3} \left( 1461582685428624t^{16} - 7497300308436576t^{15} + 18744897628849788t^{14} - 20673644844822852t^{13} + 2721598850295064t^{12} + 26159120832599697t^{11} - 36275384829659792t^{10} + 21764557558458576t^{9} - 5789999127327776t^{8} - 77229880842090t^{7} + 496393930625432t^{6} - 102436510751280t^{5} + 6025355550128t^{4} - 270833456760t^{3} - 4910191888t^{2} - 395130744t - 2379456 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ7P7

|λ| = 8.762203 λ ≈ 8.762203; multiplicity 1; Jordan blocks 1 λ ≈ -6.381102 - 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ -6.381102 + 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ 1.618898 - 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ 1.618898 + 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ 1; multiplicity 1; Jordan blocks 1 λ ≈ 0.7622032; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 2 2
Sector dimension
9
Spectral radius
$8.76220316$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.762203$11
$-6.381102 - 4.124189\mathrm{i}$11
$-6.381102 + 4.124189\mathrm{i}$11
$1.618898 - 4.124189\mathrm{i}$11
$1.618898 + 4.124189\mathrm{i}$11
$1$11
$0.7622032$11
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda - 1\right) \left(\lambda^{3} + 12 \lambda^{2} + 48 \lambda - 44\right) \left(\lambda^{3} - 12 \lambda^{2} + 48 \lambda - 172\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda - 1\right) \left(\lambda^{3} + 12 \lambda^{2} + 48 \lambda - 44\right) \left(\lambda^{3} - 12 \lambda^{2} + 48 \lambda - 172\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 6 & 6 & 6 & 0 & 0 & 40 \\ 1 & 0 & 0 & 4 & 0 & 8 & -12 & -6 & 0 \\ 2 & 0 & 0 & 0 & 0 & 4 & 18 & 12 & 0 \\ 0 & 2 & 1 & 0 & -2 & 0 & 4 & 0 & -12 \\ 0 & 1 & 0 & 0 & 1 & 0 & 8 & 4 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 12 \\ 0 & 0 & 0 & 2 & 0 & 5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & -2 & 0 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$