Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-50-17-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 + w4) + E^(w1 + 2w4)
Bundle
O_1(1) box O_2(1) + O_1(1) box O_2(2)
ZL62 label
2031.bx321
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 17
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 50
$\chi(X, \mathrm{T}_X)$ = -47

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 50
$\mathrm{c}_1^2\mathrm{c}_2$ = 92
$\mathrm{c}_2^2$ = 168
$\mathrm{c}_1\mathrm{c}_3$ = -16
$\mathrm{c}_4$ = 86
Hodge diamond
1
00
0 2 0
0 0 0 0
0 7 66 7 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
$\mathrm{HH}^{ 2 }$ = x0 + 47 0 47 x0
$\mathrm{HH}^{ 3 }$ = x0 + 3 0 0 x0 + 2 1
$\mathrm{HH}^{ 4 }$ = 41 0 0 24 0 17
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 12t + 24t^{2} + 12t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(702)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(699)

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) \left( 5t + 1 \right) ^{4} \left( 36080t^{4} + 8896t^{3} + 520t^{2} + 32t - 1 \right) \left( 287585601182000t^{10} + 378860788424800t^{9} + 208407683140380t^{8} + 62202542758080t^{7} + 11015449135758t^{6} + 1209707483014t^{5} + 87152789191t^{4} + 4455091176t^{3} + 141745504t^{2} + 800058t + 775 \right)D^{6} + \left( 5t + 1 \right) ^{3} \left( 4357957166071555200000t^{16} + 8098336054566156960000t^{15} + 6682169704563914528000t^{14} + 3232104379045471150400t^{13} + 1020297313099757744320t^{12} + 222002299554256844880t^{11} + 34422244158511317936t^{10} + 3902573397303846876t^{9} + 331528308166701280t^{8} + 21519604445877226t^{7} + 1065317025086288t^{6} + 38656859333629t^{5} + 972799801258t^{4} + 20189982800t^{3} + 554021440t^{2} + 2400949t + 1550 \right)D^{5} + \left( 5t + 1 \right) ^{2} \left( 181581548586314800000000t^{17} + 351134933755905420000000t^{16} + 303304527326136850360000t^{15} + 154672040283319475456000t^{14} + 51904985809050678117200t^{13} + 12118601144295032297280t^{12} + 2036609048037728461760t^{11} + 252772666186539520204t^{10} + 23714205196189475232t^{9} + 1708675734081398226t^{8} + 93297331104773010t^{7} + 3576243836647411t^{6} + 79818301831206t^{5} + 542588825989t^{4} - 13492052818t^{3} - 461017055t^{2} - 1721574t - 775 \right)D^{4} + t \left( 5t + 1 \right) \left( 3813212520312610800000000t^{17} + 7763377372059903660000000t^{16} + 7106865580074758498000000t^{15} + 3870741496995362780440000t^{14} + 1399770485225591454044000t^{13} + 355723017276646407026800t^{12} + 65758807228714096748720t^{11} + 9067577252940989733060t^{10} + 953190700665795824456t^{9} + 77709915373822347826t^{8} + 4912289369701341492t^{7} + 231589246866917005t^{6} + 7412816654197156t^{5} + 137152010334103t^{4} + 1156853101436t^{3} + 4836917095t^{2} + 2621540t + 5983 \right)D^{3} + 2 t ^{2} \left( 21063459636012516800000000t^{17} + 45519323446542870320000000t^{16} + 44527310139755635274500000t^{15} + 26113444223207254127100000t^{14} + 10256524273993218221035000t^{13} + 2857882390723036820293000t^{12} + 584958104978346487495200t^{11} + 90122488305824896598040t^{10} + 10661863672742125912040t^{9} + 984530323300734828452t^{8} + 71372058808110580806t^{7} + 3981846229189238561t^{6} + 160861666591242803t^{5} + 4211770482331870t^{4} + 60552650423524t^{3} + 434295793949t^{2} + 3156441683t + 5165200 \right)D^{2} + 8 t ^{2} \left( 5719818780468916200000000t^{17} + 12047072090849426730000000t^{16} + 11450748518057170511750000t^{15} + 6504824383827232556150000t^{14} + 2467070123027759900642500t^{13} + 661907774569761552088500t^{12} + 130176706256052923067100t^{11} + 19256778856275958213630t^{10} + 2189581392404551192990t^{9} + 194597079962865214001t^{8} + 13539725822505267595t^{7} + 715259214012153393t^{6} + 26567959356631785t^{5} + 608355991071969t^{4} + 7092986952185t^{3} + 42919570833t^{2} + 340848773t + 699050 \right)D + 48 t ^{2} \left( 389103318399246000000000t^{17} + 805107766659645900000000t^{16} + 749998655061029819500000t^{15} + 416506898711029666900000t^{14} + 154033986836317749335000t^{13} + 40202011149489127225000t^{12} + 7677536831144505527100t^{11} + 1102119900701729695000t^{10} + 121692355080848342350t^{9} + 10509140299649371520t^{8} + 708007011288076520t^{7} + 35750626727370783t^{6} + 1236582770931015t^{5} + 25210905463778t^{4} + 243117073258t^{3} + 1235255907t^{2} + 7407605t + 15500 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 4)

|λ| = 47.30762 λ ≈ 47.30762; multiplicity 1; Jordan blocks 1 λ ≈ -5.155707 - 11.228i; multiplicity 1; Jordan blocks 1 λ ≈ -5.155707 + 11.228i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 1, 3 λ ≈ -4.99621; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 2; Jordan blocks 1, 1 4 2
Sector dimension
10
Spectral radius
$47.307623$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$47.30762$11
$-5.155707 - 11.228\mathrm{i}$11
$-5.155707 + 11.228\mathrm{i}$11
$-5$41, 3
$-4.99621$11
$-4$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{4} \left(\lambda^{4} - 32 \lambda^{3} - 520 \lambda^{2} - 8896 \lambda - 36080\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right)^{3} \left(\lambda^{4} - 32 \lambda^{3} - 520 \lambda^{2} - 8896 \lambda - 36080\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 40 & 80 & 1116 & 384 & 729 & 729 & 9740 & 8208 & 54840 \\ 1 & -1 & 10 & 84 & 24 & 70 & 70 & 867 & 756 & 5376 \\ 1 & 3 & 4 & 112 & 48 & 66 & 66 & 1098 & 960 & 6156 \\ 0 & 1 & 1 & 3 & 0 & 8 & 8 & 54 & 40 & 405 \\ 0 & 1 & 0 & 10 & -1 & 4 & 4 & 64 & 48 & 342 \\ 0 & 0 & 1 & 3 & 2 & -2 & 2 & 48 & 48 & 288 \\ 0 & 0 & 1 & 3 & 2 & 2 & -2 & 48 & 48 & 288 \\ 0 & 0 & 0 & 2 & 0 & \frac{5}{2} & \frac{5}{2} & \frac{7}{2} & 6 & 60 \\ 0 & 0 & 0 & 1 & 1 & -\frac{7}{8} & -\frac{7}{8} & \frac{43}{8} & -\frac{1}{2} & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{3} & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$