Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-74-22-a

·

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
E^(2w4) + E^(w3 + w4) + E^(2w1 + w4)
Bundle
O_2(2) + Q_1 box O_2(1) + O_1(2) box O_2(1)
ZL62 label
30600.CwFx327
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 22
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 74
$\chi(X, \mathrm{T}_X)$ = -40

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 74
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 150
$\mathrm{c}_1\mathrm{c}_3$ = -4
$\mathrm{c}_4$ = 68
Hodge diamond
1
00
0 2 0
0 0 0 0
0 5 52 5 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 + 40 0 x0 + 40 x1
$\mathrm{HH}^{ 3 }$ = x1 + 10 0 0 x1 + 4 6
$\mathrm{HH}^{ 4 }$ = 37 0 0 15 0 22
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 17t + 38t^{2} + 17t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(647)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(655)

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( 6t + 1 \right) \left( 2t + 1 \right) ^{4} \left( 9488t^{4} + 5936t^{3} + 480t^{2} + 4t - 1 \right) \left( 1641521654784t^{10} + 3265999535616t^{9} + 2800809138176t^{8} + 1296379319424t^{7} + 333596228672t^{6} + 45395967456t^{5} + 2919379504t^{4} + 90497880t^{3} + 8052416t^{2} + 113148t + 185 \right)D^{6} + 2 \left( 2t + 1 \right) ^{3} \left( 1962419440034414592t^{16} + 5722850529647640576t^{15} + 7491676811477753856t^{14} + 5747478067142029312t^{13} + 2828661740646119424t^{12} + 922351177806871552t^{11} + 200073097581146624t^{10} + 28500505137719040t^{9} + 2627419370330752t^{8} + 166661216085824t^{7} + 9985251092704t^{6} + 618924573392t^{5} + 21643333736t^{4} + 303097652t^{3} + 15134374t^{2} + 167317t + 185 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 65413981334480486400t^{17} + 204331461914988380160t^{16} + 289043191305886187520t^{15} + 242245745467787911168t^{14} + 132136449833838530560t^{13} + 48668843697857671168t^{12} + 12218230394832854016t^{11} + 2075471300061430784t^{10} + 234910566475895808t^{9} + 17871506945343744t^{8} + 1059057936615104t^{7} + 62425336449184t^{6} + 2651848568784t^{5} + 38454514384t^{4} + 210020388t^{3} - 27805744t^{2} - 258636t - 185 \right)D^{4} + 4 t \left( 2t + 1 \right) \left( 137369360802409021440t^{17} + 466204348850844303360t^{16} + 721260258572072140800t^{15} + 667206966393186861056t^{14} + 406899899923207299072t^{13} + 170475899523093545984t^{12} + 49775492716563096576t^{11} + 10112129809552601344t^{10} + 1414256131159631872t^{9} + 135853379931883008t^{8} + 9619754413260992t^{7} + 627938028361696t^{6} + 38834824678240t^{5} + 1455712807460t^{4} + 23988163468t^{3} + 195600501t^{2} + 375697t + 130 \right)D^{3} + 8 t ^{2} \left( 303520873391989456896t^{17} + 1128470890547375898624t^{16} + 1922409809543760961536t^{15} + 1973471447035865280512t^{14} + 1350599003085155422208t^{13} + 644617476707566499840t^{12} + 218597649247721775616t^{11} + 52832740709652452608t^{10} + 9041167388688626176t^{9} + 1087727145506770432t^{8} + 94142562844419360t^{7} + 6651151684806480t^{6} + 454292216315664t^{5} + 25157148503800t^{4} + 747606171150t^{3} + 10377986627t^{2} + 73578272t + 180153 \right)D^{2} + 16 t ^{2} \left( 164843232962890825728t^{17} + 591631650783628886016t^{16} + 972669801334700384256t^{15} + 961935161939674279936t^{14} + 632144369623709931520t^{13} + 288373405259165737984t^{12} + 92932464763359446528t^{11} + 21208306093455812352t^{10} + 3407419016036432384t^{9} + 385037288205435520t^{8} + 32062295165053280t^{7} + 2314848771109456t^{6} + 160863797382648t^{5} + 8084561456740t^{4} + 202375209810t^{3} + 2525454569t^{2} + 14239403t + 36815 \right)D + 32 t ^{2} \left( 33641476114875678720t^{17} + 117811987954744688640t^{16} + 188890256251315077120t^{15} + 181869298069398355968t^{14} + 116019423828969424896t^{13} + 51169389768500798464t^{12} + 15863084871562739712t^{11} + 3463111908437063680t^{10} + 529737535196652288t^{9} + 57089216020345472t^{8} + 4645890086929216t^{7} + 342991471941440t^{6} + 23696289435912t^{5} + 1076625053508t^{4} + 23406942352t^{3} + 268819948t^{2} + 1129660t + 2405 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ6P6

|λ| = 28.528 λ ≈ 28.528; multiplicity 1; Jordan blocks 1 λ ≈ -11.32569 - 6.996813i; multiplicity 1; Jordan blocks 1 λ ≈ -11.32569 + 6.996813i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ -1.876611; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$28.5279952$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28.528$11
$-11.32569 - 6.996813\mathrm{i}$11
$-11.32569 + 6.996813\mathrm{i}$11
$-6$11
$-2$41, 3
$-1.876611$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda + 6\right) \left(\lambda^{4} - 4 \lambda^{3} - 480 \lambda^{2} - 5936 \lambda - 9488\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 6\right) \left(\lambda^{4} - 4 \lambda^{3} - 480 \lambda^{2} - 5936 \lambda - 9488\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 28 & 288 & 132 & 360 & 2400 & 1632 & 8160 \\ 1 & -2 & 6 & 28 & 28 & 36 & 468 & 324 & 1152 \\ 1 & 2 & 0 & 60 & 32 & 60 & 396 & 228 & 1248 \\ 0 & 1 & 1 & -2 & -1 & 6 & 36 & 40 & 180 \\ 0 & 1 & 0 & 6 & -2 & 6 & 12 & 4 & 108 \\ 0 & 0 & 1 & 2 & 3 & -2 & 48 & 28 & 72 \\ 0 & 0 & 0 & 1 & -\frac{1}{6} & 2 & -3 & 1 & 22 \\ 0 & 0 & 0 & 1 & \frac{3}{2} & 0 & 9 & 1 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$