Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-136-34-a

·

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

Defining data
Ambient
Fl(1, 2, 3) × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(2w3) + E^(w1 + w3) + E^(w2 + w3)
Bundle
O_2(2) + O_1(1,0) box O_2(1) + O_1(0,1) box O_2(1)
ZL62 label
2240.8lh
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 34
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 136
$\chi(X, \mathrm{T}_X)$ = -10

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 136
$\mathrm{c}_1^2\mathrm{c}_2$ = 124
$\mathrm{c}_2^2$ = 112
$\mathrm{c}_1\mathrm{c}_3$ = 44
$\mathrm{c}_4$ = 20
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 12 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 10 0 x0 + 10 x1
$\mathrm{HH}^{ 3 }$ = x1 + 25 0 0 x1 + 13 12
$\mathrm{HH}^{ 4 }$ = 34 0 0 0 0 34
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 29t + 76t^{2} + 29t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(522)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(563)

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( 3t + 1 \right) \left( 4t + 1 \right) ^{2} \left( 16t^{2} + 1 \right) \left( 368t^{3} + 128t^{2} + 8t - 1 \right) \left( 51826220089344t^{12} + 103522404028416t^{11} + 93916927864832t^{10} + 51770343172608t^{9} + 19336152857408t^{8} + 5059813583952t^{7} + 914530357928t^{6} + 109097013072t^{5} + 8080614698t^{4} + 369560517t^{3} + 14324405t^{2} + 487926t - 6868 \right)D^{6} + \left( 4t + 1 \right) \left( 62251167912755724288t^{19} + 172325546953474572288t^{18} + 223833995664895770624t^{17} + 182367994694024822784t^{16} + 104749936523868831744t^{15} + 44967877243446083584t^{14} + 14865743543175778304t^{13} + 3840327071232001024t^{12} + 780546075392523264t^{11} + 125239697502680192t^{10} + 15953900851763776t^{9} + 1654467724191648t^{8} + 150741672940664t^{7} + 13304494592464t^{6} + 1076494712874t^{5} + 63490531712t^{4} + 2217350041t^{3} + 57908729t^{2} + 2054724t - 20604 \right)D^{5} + \left( 1684443367051037245440t^{20} + 4880404089830130057216t^{19} + 6638672646605588398080t^{18} + 5662044862686083940352t^{17} + 3400328767989163425792t^{16} + 1523384457988494802944t^{15} + 524032353333581377536t^{14} + 140094697992639284224t^{13} + 29159849986989081856t^{12} + 4698784662920830784t^{11} + 577991053481321184t^{10} + 52668075150745536t^{9} + 3283932415449320t^{8} + 95761893592716t^{7} - 6077738557404t^{6} - 1058896092854t^{5} - 73475405415t^{4} - 2790143331t^{3} - 67317235t^{2} - 1684660t + 13736 \right)D^{4} + t \left( 5785696782479649669120t^{19} + 16248635980000064962560t^{18} + 21389272038889925640192t^{17} + 17638615234771332366336t^{16} + 10230115942843328299008t^{15} + 4412676076212703903744t^{14} + 1452159086110262890496t^{13} + 367628936822901619712t^{12} + 71503316138820426496t^{11} + 10622287355454630208t^{10} + 1198614179961548800t^{9} + 103225286939093984t^{8} + 7043791160141408t^{7} + 414762664589044t^{6} + 21772029938832t^{5} + 797761292274t^{4} + 3499160042t^{3} - 967835428t^{2} - 18023743t + 46258 \right)D^{3} + 2 t \left( 5302334772804134633472t^{19} + 14553665647132551413760t^{18} + 18703567458124961415168t^{17} + 15055372589069471449088t^{16} + 8520963921226549690368t^{15} + 3579648163131638513664t^{14} + 1141721469523388827648t^{13} + 277885437469901430272t^{12} + 51425575243899686656t^{11} + 7192798687212917888t^{10} + 760112634076471248t^{9} + 61997445298871472t^{8} + 4155242685515644t^{7} + 244053475631108t^{6} + 11434201365066t^{5} + 264552930724t^{4} - 7197162947t^{3} - 542979766t^{2} - 3524542t + 6868 \right)D^{2} + 4 t ^{2} \left( 2446104715630636695552t^{18} + 6605662679857039933440t^{17} + 8346287953086610931712t^{16} + 6606033364572927885312t^{15} + 3676780411292130902016t^{14} + 1516952701001147686912t^{13} + 473377215996904625152t^{12} + 112039568548417000192t^{11} + 20017677406149056512t^{10} + 2688632431825770496t^{9} + 273652210454697408t^{8} + 21962304016527856t^{7} + 1494797556909004t^{6} + 86722810803064t^{5} + 3403588987062t^{4} + 22372739614t^{3} - 4744142621t^{2} - 162539862t - 269824 \right)D + 192 t ^{3} \left( 18309167033163448320t^{17} + 48898676661710487552t^{16} + 61074688614324535296t^{15} + 47796102401299931136t^{14} + 26309354758638047232t^{13} + 10725480858170831872t^{12} + 3297905800198558336t^{11} + 765680245837201504t^{10} + 133554030016432832t^{9} + 17477681908486904t^{8} + 1746178149171658t^{7} + 140916391432476t^{6} + 9863434961316t^{5} + 567671045360t^{4} + 19195851864t^{3} - 129088550t^{2} - 35927241t - 834462 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3) × ℙ4P4

|λ| = 16.87654 λ ≈ 16.87654; multiplicity 1; Jordan blocks 1 λ ≈ -4.438272 - 1.451602i; multiplicity 1; Jordan blocks 1 λ ≈ -4.438272 + 1.451602i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 3; Jordan blocks 1, 2 λ ≈ 0 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0; multiplicity 1; Jordan blocks 1 3 2 2 2
Sector dimension
13
Spectral radius
$16.876545$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$16.87654$11
$-4.438272 - 1.451602\mathrm{i}$11
$-4.438272 + 1.451602\mathrm{i}$11
$-4$31, 2
$0 - 4\mathrm{i}$21, 1
$0 + 4\mathrm{i}$21, 1
$-3$21, 1
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 3\right)^{2} \left(\lambda + 4\right)^{3} \left(\lambda^{2} + 16\right)^{2} \left(\lambda^{3} - 8 \lambda^{2} - 128 \lambda - 368\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 3\right) \left(\lambda + 4\right)^{2} \left(\lambda^{2} + 16\right) \left(\lambda^{3} - 8 \lambda^{2} - 128 \lambda - 368\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 12 & 12 & 36 & 12 & 36 & 36 & 48 & 80 & 144 & 200 \\ 1 & -1 & 1 & 2 & 4 & 8 & 0 & 8 & 8 & 6 & 18 & 18 & 32 \\ 1 & 1 & -1 & 2 & 0 & 8 & 4 & 8 & 8 & 6 & 6 & 30 & 32 \\ 1 & 1 & 1 & 0 & 2 & 6 & 2 & 6 & 8 & 12 & 18 & 30 & 48 \\ 0 & 1 & 2 & 0 & -1 & 2 & 0 & 4 & 2 & 4 & 8 & 8 & 12 \\ 0 & 1 & 0 & 1 & 1 & -1 & 0 & 1 & 2 & 0 & 4 & 4 & 6 \\ 0 & 2 & 1 & 0 & 0 & 4 & -1 & 2 & 2 & 4 & 4 & 12 & 12 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & -1 & 2 & 0 & 0 & 8 & 6 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & -2 & 2 & 2 & 2 & 6 \\ 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0 & -2 & -1 & 2 & 4 & 6 \\ 0 & 0 & 0 & 0 & 1 & 0 & -\frac{1}{2} & 3 & 2 & \frac{1}{2} & -1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 & 2 & \frac{1}{2} & 1 & 2 & \frac{1}{2} & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 4 & 0\end{pmatrix}\notag\end{equation}$$