Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-176-42-b

·

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

Defining data
Ambient
ℙ3P3 × 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_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w5) + 2 copies of E^(w1 + w5)
Bundle
O_2(1) + 2 copies of O_1(1) box O_2(1)
ZL62 label
3031.4jj
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 42
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 176
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 176
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 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 + 11 0 x0 + 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 38 0 0 x1 + 7 31
$\mathrm{HH}^{ 4 }$ = 42 0 0 0 0 42
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 37t + 100t^{2} + 37t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(249)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(318)

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( 2t + 1 \right) \left( 4t + 1 \right) \left( 76t^{3} + 56t^{2} + 8t - 1 \right) \left( 148t^{3} + 72t^{2} + 4t + 1 \right) \left( 16419509966976t^{13} + 53651164279296t^{12} + 80306948182336t^{11} + 72642105154688t^{10} + 43671154969704t^{9} + 18031269068160t^{8} + 5135690589248t^{7} + 1000529654808t^{6} + 132025548224t^{5} + 11854781384t^{4} + 740306404t^{3} + 29184308t^{2} + 141905t + 2134 \right)D^{6} + \left( 25117384142762262528t^{21} + 124801968692755009536t^{20} + 291062989782105282560t^{19} + 422980014026761103360t^{18} + 427914345514304155648t^{17} + 318451247892516052224t^{16} + 179534619842984209280t^{15} + 77862297229281112064t^{14} + 26157381751235909760t^{13} + 6819980138424725056t^{12} + 1379509278535903552t^{11} + 216683000960398896t^{10} + 26672038336739384t^{9} + 2650536194715040t^{8} + 225931499454256t^{7} + 17632352516992t^{6} + 1263023151360t^{5} + 77737366672t^{4} + 3742503860t^{3} + 116863150t^{2} + 417179t + 4268 \right)D^{5} + \left( 169911716259862364160t^{21} + 809298708032697830400t^{20} + 1811860638928707708416t^{19} + 2530443839581933090304t^{18} + 2460430506616154507008t^{17} + 1757584000050920679360t^{16} + 948731722035044540896t^{15} + 392665030526427356416t^{14} + 125465211995960567104t^{13} + 31022497711741578208t^{12} + 5935194493006307072t^{11} + 877011257226591920t^{10} + 99467740639012360t^{9} + 8447820488251048t^{8} + 488908372490172t^{7} + 12093466004520t^{6} - 724359929768t^{5} - 79395379104t^{4} - 3452914804t^{3} - 104545182t^{2} - 324647t - 2134 \right)D^{4} + t \left( 583609808023005511680t^{20} + 2691023184611360655360t^{19} + 5838494835117228157952t^{18} + 7907857673452812707840t^{17} + 7454361003479071084544t^{16} + 5153844069591570061696t^{15} + 2685322739866577269312t^{14} + 1069413895831214696192t^{13} + 327882752656521030016t^{12} + 77689380510151427136t^{11} + 14264314449232234944t^{10} + 2035675868485295632t^{9} + 226274999309426600t^{8} + 19436805408133616t^{7} + 1239187316432040t^{6} + 52612340792224t^{5} + 1205553748800t^{4} + 13174152696t^{3} + 145866540t^{2} + 2293050t + 2425 \right)D^{3} + 4 t ^{2} \left( 267426266461174677504t^{19} + 1203776872397273793792t^{18} + 2551393121506828178048t^{17} + 3377229129463159266176t^{16} + 3109443716513309572416t^{15} + 2096086503644540910416t^{14} + 1062035530603387535688t^{13} + 410108406434807950688t^{12} + 121627445918697188320t^{11} + 27843451145383626312t^{10} + 4942698730717029312t^{9} + 683643364163193392t^{8} + 73772792363824752t^{7} + 6117481189725910t^{6} + 368144456657325t^{5} + 14088955223216t^{4} + 276694049728t^{3} + 3712906388t^{2} + 67125230t + 480392 \right)D^{2} + 8 t ^{2} \left( 123370680936508760064t^{19} + 545826162497405691648t^{18} + 1137546507974820770944t^{17} + 1480882576067821991296t^{16} + 1340069602642743716352t^{15} + 886385766879544605616t^{14} + 439644659177854959416t^{13} + 165777598500928355376t^{12} + 47913239495179167704t^{11} + 10680147610877668184t^{10} + 1847442480579331616t^{9} + 249447310489233848t^{8} + 26278374217304248t^{7} + 2109991077476234t^{6} + 119892987731787t^{5} + 4112438741802t^{4} + 67057323035t^{3} + 1007116824t^{2} + 18743844t + 102432 \right)D + 16 t ^{2} \left( 22162397773025525760t^{19} + 96892808109327751680t^{18} + 199592395401673130496t^{17} + 256842115595994605568t^{16} + 229607893820512806848t^{15} + 149829230077823979104t^{14} + 73173081065331534640t^{13} + 27112473607885245520t^{12} + 7687754203114987864t^{11} + 1680150594259746304t^{10} + 285128677234300992t^{9} + 37817916814404248t^{8} + 3910684468451288t^{7} + 305790337291444t^{6} + 16549961091302t^{5} + 516285437894t^{4} + 7137660583t^{3} + 116972414t^{2} + 1850760t + 8536 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 4)

|λ| = 12.8275 λ ≈ 12.8275; multiplicity 1; Jordan blocks 1 λ ≈ -0.9122573 - 8.197474i; multiplicity 1; Jordan blocks 1 λ ≈ -0.9122573 + 8.197474i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -2.41375 - 0.3139739i; multiplicity 1; Jordan blocks 1 λ ≈ -2.41375 + 0.3139739i; multiplicity 1; Jordan blocks 1 λ ≈ -2.175485; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$12.8275008$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.8275$11
$-0.9122573 - 8.197474\mathrm{i}$11
$-0.9122573 + 8.197474\mathrm{i}$11
$-4$11
$-2.41375 - 0.3139739\mathrm{i}$11
$-2.41375 + 0.3139739\mathrm{i}$11
$-2.175485$11
$-2$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 2\right) \left(\lambda + 4\right) \left(\lambda^{3} + 4 \lambda^{2} + 72 \lambda + 148\right) \left(\lambda^{3} - 8 \lambda^{2} - 56 \lambda - 76\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 2\right) \left(\lambda + 4\right) \left(\lambda^{3} + 4 \lambda^{2} + 72 \lambda + 148\right) \left(\lambda^{3} - 8 \lambda^{2} - 56 \lambda - 76\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 4 & 18 & 18 & 3 & 48 & 96 & 40 \\ 2 & 0 & 4 & 4 & 0 & 4 & 3 & 12 & 16 \\ 1 & 0 & 0 & 2 & 4 & 0 & 12 & 30 & 16 \\ 0 & 1 & 2 & 0 & 0 & 2 & 0 & 0 & 3 \\ 0 & 2 & 0 & 4 & 0 & 1 & 4 & 4 & 0 \\ 0 & 0 & 2 & 0 & 0 & -2 & 2 & 8 & 6 \\ 0 & 0 & 0 & 2 & -\frac{4}{3} & \frac{8}{3} & -\frac{8}{3} & -\frac{8}{3} & 0 \\ 0 & 0 & 0 & 2 & \frac{7}{3} & -\frac{1}{6} & \frac{8}{3} & \frac{8}{3} & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 8 & 0\end{pmatrix}\notag\end{equation}$$