Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-405-85-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × 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^(w2 + w3) + E^(w1 + w3)
Bundle
O_2(1) box O_3(1) + O_1(1) box O_3(1)
ZL62 label
1140.lh
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 85
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 405
$\chi(X, \mathrm{T}_X)$ = 12

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 405
$\mathrm{c}_1^2\mathrm{c}_2$ = 198
$\mathrm{c}_2^2$ = 95
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 12 0 12
$\mathrm{HH}^{ 2 }$ = 58 0 0 58
$\mathrm{HH}^{ 3 }$ = 120 0 0 0 120
$\mathrm{HH}^{ 4 }$ = 85 0 0 0 0 85
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 80t + 243t^{2} + 80t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(11)
Coates et al. (2020) Obro(4,86)
Coates–Kasprzyk–Prince (2015) CKP(15)

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( 13t^{2} + 5t + 1 \right) \left( 5521t^{6} + 5625t^{5} + 1350t^{4} + 54t^{3} - 2t - 1 \right) \left( 6726750968474784t^{13} + 23860049920203744t^{12} + 32318566894178224t^{11} + 23791580171951636t^{10} + 11240706326922032t^{9} + 3664846854450098t^{8} + 823188507063287t^{7} + 122420987840582t^{6} + 12852245367097t^{5} + 1190812408215t^{4} + 78383362185t^{3} + 1889433675t^{2} + 17532450t + 820125 \right)D^{6} + \left( 20277562084934308225344t^{22} + 90180650567028484311264t^{21} + 162545050551460468122400t^{20} + 160638243689782195219056t^{19} + 96547946840576595796240t^{18} + 34602510001824982823352t^{17} + 4516256952397101421110t^{16} - 2623110120975967956145t^{15} - 1967601481585360840416t^{14} - 709509253730366681995t^{13} - 171180404731976334771t^{12} - 30825729520927284237t^{11} - 4629753689467906001t^{10} - 659292333579306269t^{9} - 93116059780562570t^{8} - 11925240489960550t^{7} - 1230164193334021t^{6} - 101377721635084t^{5} - 7498405716615t^{4} - 391734743175t^{3} - 7425785700t^{2} - 44396100t - 1640250 \right)D^{5} + \left( 168979684041119235211200t^{22} + 751075595488245431832480t^{21} + 1350707329109553467292864t^{20} + 1349738948897114562909552t^{19} + 850328799629933475523636t^{18} + 351640168414258062956848t^{17} + 88236645078503676278064t^{16} + 5641739561985425968065t^{15} - 5433472067115060687698t^{14} - 2596147541851053593601t^{13} - 653533817128715166314t^{12} - 112811139762558737585t^{11} - 14547984259406389896t^{10} - 1344970093953427806t^{9} - 52787077824271395t^{8} + 7411384782457995t^{7} + 1440585043081030t^{6} + 118216408918712t^{5} + 7835278955325t^{4} + 393643496250t^{3} + 7484070600t^{2} + 29506275t + 820125 \right)D^{4} + t \left( 709714672972700787887040t^{21} + 3153099075568557420183840t^{20} + 5652296710376575628476640t^{19} + 5672075338843771532184192t^{18} + 3661460558617673350288064t^{17} + 1616230594562951049499080t^{16} + 482920895087826779431230t^{15} + 83499365889699364025113t^{14} + 322513877724360839160t^{13} - 4175245807983919852585t^{12} - 1284176612392129427547t^{11} - 242411610653306368203t^{10} - 35493858649818856943t^{9} - 4160893652412669217t^{8} - 358800501469526500t^{7} - 20422359736265266t^{6} - 711896770521509t^{5} - 16901756740424t^{4} - 351088069995t^{3} - 7162183305t^{2} - 1456650t - 182250 \right)D^{3} + 2 t ^{2} \left( 784065733950793251379968t^{20} + 3482224833375415686353184t^{19} + 6221995558871861914870400t^{18} + 6247818038621425704249724t^{17} + 4084404261203239151377688t^{16} + 1864084445861788399440082t^{15} + 599481782835192745401152t^{14} + 127265073076937758686433t^{13} + 13703148401922763890516t^{12} - 806793563589643392282t^{11} - 605659148958549249585t^{10} - 136518110536992510855t^{9} - 22244507075523892776t^{8} - 2729014216880754987t^{7} - 230125217966364634t^{6} - 12377959025564579t^{5} - 426882009382816t^{4} - 11960047624065t^{3} - 310358617215t^{2} - 5032882125t + 275400 \right)D^{2} + 36 t ^{3} \left( 47314311531513385859136t^{19} + 210081239149336735151296t^{18} + 374320196029606789903200t^{17} + 375549274975298855873956t^{16} + 247104384313487904920768t^{15} + 114829232163851027583920t^{14} + 38294357174406297215558t^{13} + 8817721294363939443588t^{12} + 1258717653978329803826t^{11} + 72860435020553286088t^{10} - 11096409423214660442t^{9} - 4208151996051895799t^{8} - 822287035916842516t^{7} - 106731583017104058t^{6} - 8774635752736341t^{5} - 444475631723184t^{4} - 15147405717444t^{3} - 465695685045t^{2} - 12732134445t - 175670775 \right)D + 72 t ^{3} \left( 9655981945206813440640t^{19} + 42866412311026908464640t^{18} + 76221189610876395287488t^{17} + 76383816211763381576360t^{16} + 50424166146950541771232t^{15} + 23665382869903995811574t^{14} + 8044049013443940702508t^{13} + 1924706912315520322850t^{12} + 304474620899821332202t^{11} + 28184609439406476755t^{10} + 489221629910975128t^{9} - 428305938916840449t^{8} - 112260709227556623t^{7} - 15560995541488134t^{6} - 1252180280553525t^{5} - 60276657975345t^{4} - 2063569148865t^{3} - 68161123350t^{2} - 1873657575t - 22143375 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 6.736275 λ ≈ 6.736275; multiplicity 1; Jordan blocks 1 λ ≈ 0.219865 - 6.006624i; multiplicity 1; Jordan blocks 1 λ ≈ 0.219865 + 6.006624i; multiplicity 1; Jordan blocks 1 λ ≈ -3.85261 - 0.7627077i; multiplicity 1; Jordan blocks 1 λ ≈ -3.85261 + 0.7627077i; multiplicity 1; Jordan blocks 1 λ ≈ -2.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.470786; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
12
Spectral radius
$6.73627548$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.736275$11
$0.219865 - 6.006624\mathrm{i}$11
$0.219865 + 6.006624\mathrm{i}$11
$-3.85261 - 0.7627077\mathrm{i}$11
$-3.85261 + 0.7627077\mathrm{i}$11
$-2.5 - 2.598076\mathrm{i}$21, 1
$-2.5 + 2.598076\mathrm{i}$21, 1
$2$21, 1
$-1.470786$11

Characteristic polynomial

$$\begin{equation}\left(\lambda - 2\right)^{2} \left(\lambda^{2} + 5 \lambda + 13\right)^{2} \left(\lambda^{6} + 2 \lambda^{5} - 54 \lambda^{3} - 1350 \lambda^{2} - 5625 \lambda - 5521\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda - 2\right) \left(\lambda^{2} + 5 \lambda + 13\right) \left(\lambda^{6} + 2 \lambda^{5} - 54 \lambda^{3} - 1350 \lambda^{2} - 5625 \lambda - 5521\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 4 & 4 & 5 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 4 \\ 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 \\ 3 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 3 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 & 1 & -3 & -2 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 4 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 4 & 0\end{pmatrix}\notag\end{equation}$$