Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-100-27-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 6)
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_7ω1ω2ω3ω4ω5ω6ω7
Bundle
4 copies of E^(w4) + E^(w1 + w3)
Bundle
4 copies of O_2(1) + O_1(1) box U_2^v
ZL62 label
2051.3333w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 100
$\chi(X, \mathrm{T}_X)$ = -18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 100
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 122
$\mathrm{c}_1\mathrm{c}_3$ = 28
$\mathrm{c}_4$ = 22
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 16 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 + x2 + 18 0 x0 + 18 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 9 0 0 x1 - x2 - 18 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 54t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(603)
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) \left( 3t + 1 \right) ^{2} \left( 189624t^{7} + 168064t^{6} + 61382t^{5} + 12159t^{4} + 1658t^{3} + 194t^{2} + 9t - 1 \right) \left( 284782749346567296t^{16} + 828619723050697728t^{15} + 1175680259087317632t^{14} + 1106945630834333376t^{13} + 776821238475575688t^{12} + 421121469782495664t^{11} + 175997501549757248t^{10} + 55925162148419532t^{9} + 13304353146365424t^{8} + 2331613904014606t^{7} + 295763664737778t^{6} + 26728194292633t^{5} + 1722445215312t^{4} + 82636192176t^{3} + 3013207650t^{2} + 56185538t - 740400 \right)D^{6} + 2 \left( 3t + 1 \right) \left( 6804207151823778094024704t^{25} + 28403637523805372426408448t^{24} + 57848780526502358287908864t^{23} + 77278335242466177980876160t^{22} + 76399303776361125678768960t^{21} + 59328264271255390616915808t^{20} + 37240702994350022808135744t^{19} + 19126751603894426625391128t^{18} + 8072748922368178916045628t^{17} + 2803582390351286130188964t^{16} + 801347044328976228925328t^{15} + 188476396552036755854676t^{14} + 36457509343780941197420t^{13} + 5795843739421266020404t^{12} + 757401583768965386127t^{11} + 81694835430173985623t^{10} + 7383353996249484210t^{9} + 579480580235476709t^{8} + 41557355800277195t^{7} + 2766121456986490t^{6} + 159726548880580t^{5} + 7208461308947t^{4} + 244602307538t^{3} + 6720198897t^{2} + 116073076t - 1110600 \right)D^{5} + \left( 340210357591188904701235200t^{26} + 1477826907806592158023480320t^{25} + 3134249812673624996590427136t^{24} + 4363440863817260634966888192t^{23} + 4499938640368172627319701376t^{22} + 3649506924693852887042924352t^{21} + 2395723851869095835074407360t^{20} + 1288636609339336713413004720t^{19} + 570393016839343884367821960t^{18} + 207981085815606954801849536t^{17} + 62461278528544863305127504t^{16} + 15437905031551622561673268t^{15} + 3135678219031161154373756t^{14} + 522266669893610102094570t^{13} + 71117374222581503528908t^{12} + 7886343066864901127487t^{11} + 707978444766683330110t^{10} + 50804900788533083713t^{9} + 2808568762040854062t^{8} + 104833597942719826t^{7} + 932934956015129t^{6} - 194462261329733t^{5} - 15205648588972t^{4} - 598295320180t^{3} - 14531752868t^{2} - 188532890t + 1480800 \right)D^{4} + t \left( 1428883501882993399745187840t^{25} + 6022870223771246939647349760t^{24} + 12429570811899744634859228160t^{23} + 16909317743286461855339840256t^{22} + 17107458113315893844594044800t^{21} + 13634436820798939645640886336t^{20} + 8783253697832029007976825792t^{19} + 4617470677102216516454405040t^{18} + 1986309199763173254847465992t^{17} + 699408118558853752001531856t^{16} + 201509029068843396608319000t^{15} + 47467869699823147635889928t^{14} + 9131007022049856174678360t^{13} + 1432196518229874880292916t^{12} + 182960181585065300387374t^{11} + 19045698631162283750732t^{10} + 1621649804634442429958t^{9} + 113713852692997506682t^{8} + 6579965247887609216t^{7} + 305684452161518043t^{6} + 10216460247795224t^{5} + 159977132800208t^{4} - 3944419341066t^{3} - 245426915086t^{2} - 3198873138t + 7544676 \right)D^{3} + t \left( 3157152118446233035627462656t^{25} + 12996622827281058742657953792t^{24} + 26259285620163146384280637440t^{23} + 35104088234328475876583030784t^{22} + 35013125564458062773145555456t^{21} + 27539587958910997053935856384t^{20} + 17479037690826491151508677888t^{19} + 9019112496066785822010497856t^{18} + 3789571012185747365540036256t^{17} + 1296574342748389838558124224t^{16} + 361114107262287458269851744t^{15} + 81826527400013154319778368t^{14} + 15073310752768487847854072t^{13} + 2255813627533674956399360t^{12} + 274385307189902095703236t^{11} + 27205897462359096387644t^{10} + 2212410891697218410212t^{9} + 148280227578446393006t^{8} + 8084434130477657263t^{7} + 335481171960814670t^{6} + 8540452056464000t^{5} + 2390233364940t^{4} - 8283223352064t^{3} - 232054717476t^{2} - 1242974000t + 2221200 \right)D^{2} + 6 t ^{2} \left( 571553400753197359898075136t^{24} + 2311570852908871437346904064t^{23} + 4597502042872861705449400320t^{22} + 6067926464764845582689822208t^{21} + 5990094274078633767760606464t^{20} + 4665878095836364376095516032t^{19} + 2927854112727350177536892544t^{18} + 1488838566583284731817911712t^{17} + 614050001236184549184672624t^{16} + 205375021263443518509556896t^{15} + 55691342270521102778273040t^{14} + 12240760371552436152836016t^{13} + 2180186687585612764025616t^{12} + 314756676994931624498112t^{11} + 36911992058707216211616t^{10} + 3534882499226301985808t^{9} + 278532869279347544664t^{8} + 18067394561021171770t^{7} + 935117350837312741t^{6} + 34489912732539344t^{5} + 588693121735952t^{4} - 16650879525608t^{3} - 1189984080132t^{2} - 21709115616t - 31174400 \right)D + 144 t ^{3} \left( 9720295931176825848606720t^{23} + 38838293238976786648212480t^{22} + 76416489843428071837817856t^{21} + 99982335241829476927800576t^{20} + 98011752147086082002340480t^{19} + 75833778627965014377110208t^{18} + 47203916840834522086791744t^{17} + 23752234594743791320189296t^{16} + 9664932017956735864441464t^{15} + 3179364205279575449103952t^{14} + 845412604487708980148208t^{13} + 181701089526949175365508t^{12} + 31570008609760377956046t^{11} + 4439371527575699723932t^{10} + 507075503585704059804t^{9} + 47393154958024027516t^{8} + 3655366197962839378t^{7} + 231761162290673614t^{6} + 11527512901491037t^{5} + 385250732297088t^{4} + 3908967628872t^{3} - 324386902624t^{2} - 15466451076t - 219898800 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 6)

|λ| = 22.3585 λ ≈ 22.3585; multiplicity 1; Jordan blocks 1 λ ≈ 0.4927756 - 6.173351i; multiplicity 1; Jordan blocks 1 λ ≈ 0.4927756 + 6.173351i; multiplicity 1; Jordan blocks 1 λ ≈ -4.054258 - 0.2170487i; multiplicity 1; Jordan blocks 1 λ ≈ -4.054258 + 0.2170487i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -3.11777 - 1.922061i; multiplicity 1; Jordan blocks 1 λ ≈ -3.11777 + 1.922061i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 2 2
Sector dimension
10
Spectral radius
$22.3585047$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22.3585$11
$0.4927756 - 6.173351\mathrm{i}$11
$0.4927756 + 6.173351\mathrm{i}$11
$-4.054258 - 0.2170487\mathrm{i}$11
$-4.054258 + 0.2170487\mathrm{i}$11
$-4$11
$-3.11777 - 1.922061\mathrm{i}$11
$-3.11777 + 1.922061\mathrm{i}$11
$-3$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda + 4\right) \left(\lambda^{7} - 9 \lambda^{6} - 194 \lambda^{5} - 1658 \lambda^{4} - 12159 \lambda^{3} - 61382 \lambda^{2} - 168064 \lambda - 189624\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda + 4\right) \left(\lambda^{7} - 9 \lambda^{6} - 194 \lambda^{5} - 1658 \lambda^{4} - 12159 \lambda^{3} - 61382 \lambda^{2} - 168064 \lambda - 189624\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 28 & 126 & 33 & 93 & 162 & 504 & 876 & 2150 \\ 1 & -1 & 4 & 20 & 0 & 20 & 36 & 60 & 99 & 260 \\ 1 & 1 & 2 & 18 & 6 & 14 & 24 & 87 & 153 & 400 \\ 0 & 1 & 1 & 1 & 0 & 3 & 6 & 8 & 12 & 30 \\ 0 & 1 & 0 & 4 & -1 & 0 & 0 & 20 & 36 & 75 \\ 0 & 0 & 1 & 1 & 1 & -3 & 0 & 0 & 0 & 15 \\ 0 & 0 & 1 & 1 & 0 & 2 & 0 & 10 & 18 & 45 \\ 0 & 0 & 0 & -4 & -5 & 11 & 18 & -9 & -12 & -20 \\ 0 & 0 & 0 & \frac{13}{3} & \frac{10}{3} & -5 & -8 & 7 & 10 & \frac{80}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{5} & \frac{12}{5} & 0\end{pmatrix}\notag\end{equation}$$