Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-116-30-b

·

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

Defining data
Ambient
Fl(1, 3, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
3 copies of E^(w3) + E^(2w1)
Bundle
3 copies of O(0,1) + O(2,0)
ZL62 label
44.2224
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 116
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 116
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 116
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 26 0 0 x1 + 19 7
$\mathrm{HH}^{ 4 }$ = 30 0 0 0 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 64t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(582)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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) ^{2} \left( 4528t^{6} + 1248t^{5} + 3184t^{4} + 1496t^{3} + 216t^{2} + 4t - 1 \right) \left( 1620052518845200t^{13} + 2730547640901200t^{12} + 2196090643690944t^{11} + 1123662104360556t^{10} + 400018814735832t^{9} + 99465023771564t^{8} + 17369414497028t^{7} + 2445494232712t^{6} + 355822107280t^{5} + 47406799231t^{4} + 3938610794t^{3} + 139570780t^{2} - 251428t + 8932 \right)D^{6} + 2 \left( 4t + 1 \right) \left( 249410325381256230400t^{20} + 530678764103501654400t^{19} + 633007009688546415616t^{18} + 551350308864100374656t^{17} + 369677801077057885760t^{16} + 193532756591998477824t^{15} + 80029781840452604736t^{14} + 26117666340417956440t^{13} + 6675034124950643600t^{12} + 1361705534508826144t^{11} + 241004201027475016t^{10} + 40931199610577764t^{9} + 6419349278721804t^{8} + 793700968392314t^{7} + 68102266665664t^{6} + 3904881028408t^{5} + 177514103485t^{4} + 8696918905t^{3} + 287655480t^{2} - 448598t + 8932 \right)D^{5} + \left( 13497499961809160704000t^{21} + 31261274161764890622400t^{20} + 37392404146322765036992t^{19} + 30689646144146605039296t^{18} + 19057772727633958858896t^{17} + 9345803595387415564320t^{16} + 3709496809610917918624t^{15} + 1200801903066398650784t^{14} + 315595992064739363584t^{13} + 68285136447823976656t^{12} + 13000212240033203716t^{11} + 2360377642012175092t^{10} + 396509289873516004t^{9} + 53846627082472196t^{8} + 5190458454615892t^{7} + 313094044596460t^{6} + 8667012897346t^{5} - 161546243555t^{4} - 20255265206t^{3} - 578850684t^{2} - 340756t - 8932 \right)D^{4} + 2 t \left( 23180489064846167296000t^{20} + 52668144651850300792000t^{19} + 60078638250378220595776t^{18} + 45804837800116151676480t^{17} + 25932656191264089556176t^{16} + 11462302187108628389664t^{15} + 4100022015603934733568t^{14} + 1216846665705096979392t^{13} + 304074744567842525544t^{12} + 66266582106747672304t^{11} + 13517601095628082612t^{10} + 2633439822108744172t^{9} + 443743248682707720t^{8} + 56724907923892472t^{7} + 4971305523839534t^{6} + 269926564606984t^{5} + 7644963972594t^{4} + 82682084426t^{3} + 1438667643t^{2} + 31345880t + 10934 \right)D^{3} + 2 t ^{2} \left( 42487782488477531955200t^{19} + 95261290903256906084000t^{18} + 105714180347818059676960t^{17} + 77390234018572025511776t^{16} + 41537128670854597974808t^{15} + 17137733185241077366224t^{14} + 5657166920776574573472t^{13} + 1563736474918882041328t^{12} + 378423772406128970616t^{11} + 84339506588913882216t^{10} + 18017977531018771478t^{9} + 3539225397402491512t^{8} + 569227645334114790t^{7} + 67159861334560680t^{6} + 5326170604278508t^{5} + 255573487182624t^{4} + 6185517235703t^{3} + 73401018390t^{2} + 2314930892t + 21674948 \right)D^{2} + 8 t ^{2} \left( 9800358667922303641600t^{19} + 21782378945787842998000t^{18} + 23819111667023005029808t^{17} + 17096278480997307880608t^{16} + 8935808011059523187252t^{15} + 3543878460670248311416t^{14} + 1108910778786280344732t^{13} + 291715438289041567752t^{12} + 69573119276126521096t^{11} + 15893016252927960668t^{10} + 3461697149174501123t^{9} + 663943991407095404t^{8} + 100633580664201868t^{7} + 11004478048292540t^{6} + 800445618398684t^{5} + 34588028982280t^{4} + 736726760476t^{3} + 9992208752t^{2} + 333957504t + 1929312 \right)D + 16 t ^{2} \left( 1760543473279455744000t^{19} + 3891678684918013053600t^{18} + 4224764945939487407520t^{17} + 3007562490802847479920t^{16} + 1554721011012879163656t^{15} + 604443700451463984480t^{14} + 183322510486699614844t^{13} + 46817536130360479584t^{12} + 11119099882680180576t^{11} + 2583289464765683840t^{10} + 562393126938697830t^{9} + 104143010824423532t^{8} + 14910843132645127t^{7} + 1527558963158478t^{6} + 103584006841447t^{5} + 4130195895560t^{4} + 82293005284t^{3} + 1383168744t^{2} + 37430316t + 142912 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 19.47591 λ ≈ 19.47591; multiplicity 1; Jordan blocks 1 λ ≈ -5.627086 - 2.460671i; multiplicity 1; Jordan blocks 1 λ ≈ -5.627086 + 2.460671i; multiplicity 1; Jordan blocks 1 λ ≈ -4.488427; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ 0.1333442 - 1.164253i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1333442 + 1.164253i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1 2
Sector dimension
9
Spectral radius
$19.4759096$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.47591$11
$-5.627086 - 2.460671\mathrm{i}$11
$-5.627086 + 2.460671\mathrm{i}$11
$-4.488427$11
$-4$22
$0.1333442 - 1.164253\mathrm{i}$11
$0.1333442 + 1.164253\mathrm{i}$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right)^{2} \left(\lambda^{6} - 4 \lambda^{5} - 216 \lambda^{4} - 1496 \lambda^{3} - 3184 \lambda^{2} - 1248 \lambda - 4528\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right)^{2} \left(\lambda^{6} - 4 \lambda^{5} - 216 \lambda^{4} - 1496 \lambda^{3} - 3184 \lambda^{2} - 1248 \lambda - 4528\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 108 & 60 & 54 & 64 & 256 & 340 \\ 1 & 0 & 3 & 16 & 12 & 6 & 12 & 48 & 64 \\ 1 & 2 & 0 & 24 & 12 & 12 & 12 & 48 & 64 \\ 0 & 1 & 1 & 0 & 2 & 3 & 0 & 8 & 12 \\ 0 & 1 & 0 & 3 & -2 & 0 & 2 & 4 & 6 \\ 0 & 0 & \frac{5}{3} & \frac{10}{3} & \frac{4}{3} & -2 & 4 & \frac{32}{3} & 12 \\ 0 & 0 & 0 & -3 & -5 & 3 & -3 & -2 & 0 \\ 0 & 0 & 0 & 5 & \frac{7}{2} & \frac{3}{2} & \frac{3}{2} & 3 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 & 0\end{pmatrix}\notag\end{equation}$$