Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-55-18-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w2 + w3) + E^(w2 + 2w3)
Bundle
Q_1 box O_2(1) + Q_1 box O_2(2)
ZL62 label
20600.w7x371
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 55
$\chi(X, \mathrm{T}_X)$ = -40

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 55
$\mathrm{c}_1^2\mathrm{c}_2$ = 94
$\mathrm{c}_2^2$ = 156
$\mathrm{c}_1\mathrm{c}_3$ = -10
$\mathrm{c}_4$ = 59
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 47 3 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 + 40 0 x0 + 40 x2
$\mathrm{HH}^{ 3 }$ = x1 + 3 0 0 x1 3
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 22 0 0 x1 - x2 + 4 0 18
$\mathrm{HH}^{ 5 }$ = 1 0 0 1 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 27t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(671)
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( 3t + 1 \right) \left( 4t + 1 \right) ^{3} \left( 111952t^{5} + 71976t^{4} + 11845t^{3} + 168t^{2} + 28t - 1 \right) \left( 17212590612561920t^{13} + 27831485697269760t^{12} + 20139832110836736t^{11} + 8648823743781632t^{10} + 2531108581528320t^{9} + 567305807957168t^{8} + 109160302874496t^{7} + 19598370508076t^{6} + 3141777356740t^{5} + 368628732956t^{4} + 24904124794t^{3} + 730414907t^{2} + 3083176t + 6435 \right)D^{6} + 2 \left( 4t + 1 \right) ^{2} \left( 242799976976449040547840t^{20} + 629183210644126192762880t^{19} + 748621921342193457299456t^{18} + 543470764577685534081024t^{17} + 271024232174324565911552t^{16} + 99832769591851004416512t^{15} + 28833717323427801366272t^{14} + 6930662940753809528128t^{13} + 1466703898088979998640t^{12} + 281064147051356843312t^{11} + 47452877876698901240t^{10} + 6561197885520773198t^{9} + 683153195761970498t^{8} + 49589329258793681t^{7} + 2328315132487399t^{6} + 66246294070182t^{5} + 1458394561886t^{4} + 54975846148t^{3} + 1453241706t^{2} + 4656939t + 6435 \right)D^{5} + \left( 4t + 1 \right) \left( 16186665131763269369856000t^{21} + 42731398632211868549120000t^{20} + 52145822581440671801147392t^{19} + 39123820950218732280872960t^{18} + 20343780592243430209028096t^{17} + 7891532259919329683304448t^{16} + 2423765777798304987644928t^{15} + 623837619562186887673856t^{14} + 141543765242835337373248t^{13} + 29012344252005718989040t^{12} + 5256803402234179587616t^{11} + 793382939819077226076t^{10} + 93002709925957916104t^{9} + 7926903179356550344t^{8} + 459118346947262338t^{7} + 16479760431361517t^{6} + 301545324116669t^{5} + 284090927115t^{4} - 105770389210t^{3} - 2487778045t^{2} - 6927593t - 6435 \right)D^{4} + 2 t \left( 135967987106811462706790400t^{21} + 371414366109055974637568000t^{20} + 471802231743340646537625600t^{19} + 370912284149709144201232384t^{18} + 203549858408665577060466688t^{17} + 83929599514399284353622016t^{16} + 27552652591407697159518208t^{15} + 7591379131802008500822528t^{14} + 1837043054743323558046464t^{13} + 399772890133650600025984t^{12} + 77194420430916085240496t^{11} + 12633051936179473100704t^{10} + 1653658928827269149808t^{9} + 163725599629569896434t^{8} + 11610690252894967990t^{7} + 553404072542805179t^{6} + 16161179110610009t^{5} + 247264504813054t^{4} + 1550559664707t^{3} + 4262728163t^{2} + 23278724t + 19188 \right)D^{3} + 4 t ^{2} \left( 150212252422763139752263680t^{20} + 392133753015075080713011200t^{19} + 476325900534147426996977664t^{18} + 358571573310202052382097408t^{17} + 189128437682065079362945024t^{16} + 75516729079838469135593472t^{15} + 24284436727454633911379968t^{14} + 6641088122244082926755584t^{13} + 1609702214904480783879808t^{12} + 350276359896405822799088t^{11} + 66704848630474733549776t^{10} + 10551905979875386535500t^{9} + 1308438085362653739160t^{8} + 120472834500763474272t^{7} + 7798706653751812096t^{6} + 331490816472363920t^{5} + 8321570377983525t^{4} + 103517339815829t^{3} + 691313867399t^{2} + 5516936927t + 14983100 \right)D^{2} + 8 t ^{2} \left( 81580792264086877624074240t^{20} + 205729340773662436150476800t^{19} + 241483688312091382577627136t^{18} + 175857066591325128923414528t^{17} + 90036264060753452385992704t^{16} + 35132241485785859395702784t^{15} + 11147114584466045965164544t^{14} + 3036837990730986782661632t^{13} + 736771550177213099093888t^{12} + 159789699445952714535600t^{11} + 29949649237682994384976t^{10} + 4588924202160585318652t^{9} + 542626011646203698196t^{8} + 46951608710743233390t^{7} + 2812016799514805986t^{6} + 108262409551261075t^{5} + 2369042687382265t^{4} + 23523481871856t^{3} + 148550470799t^{2} + 1382958833t + 3389100 \right)D + 192 t ^{2} \left( 1387428439865423088844800t^{20} + 3415614255665925062656000t^{19} + 3913749008738757313560576t^{18} + 2784011769626407390363648t^{17} + 1395874209744013223190528t^{16} + 536193838376736802177024t^{15} + 168687038757012022987776t^{14} + 45864199900060925408320t^{13} + 11129828954403406912832t^{12} + 2403819186258421341744t^{11} + 444276468518204822448t^{10} + 66324774101474151520t^{9} + 7551552507957198576t^{8} + 621905807157170748t^{7} + 34992173448514476t^{6} + 1242613413334496t^{5} + 24242856760640t^{4} + 196729102845t^{3} + 1087743015t^{2} + 8635419t + 19305 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ6P6

|λ| = 40.49657 λ ≈ 40.49657; multiplicity 1; Jordan blocks 1 λ ≈ -3.164951 - 16.71727i; multiplicity 1; Jordan blocks 1 λ ≈ -3.164951 + 16.71727i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 3; Jordan blocks 3 λ ≈ -3.083333 - 0.2067334i; multiplicity 1; Jordan blocks 1 λ ≈ -3.083333 + 0.2067334i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$40.4965688$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$40.49657$11
$-3.164951 - 16.71727\mathrm{i}$11
$-3.164951 + 16.71727\mathrm{i}$11
$-4$33
$-3.083333 - 0.2067334\mathrm{i}$11
$-3.083333 + 0.2067334\mathrm{i}$11
$-3$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{3} \left(\lambda^{5} - 28 \lambda^{4} - 168 \lambda^{3} - 11845 \lambda^{2} - 71976 \lambda - 111952\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{3} \left(\lambda^{5} - 28 \lambda^{4} - 168 \lambda^{3} - 11845 \lambda^{2} - 71976 \lambda - 111952\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 56 & 672 & 336 & 1200 & 12352 & 5824 & 60960 \\ 1 & 0 & 10 & 24 & 0 & 84 & 666 & 372 & 5424 \\ 1 & 0 & 4 & 64 & 48 & 120 & 1686 & 876 & 8736 \\ 0 & 1 & 1 & 4 & -3 & 16 & 12 & -8 & 252 \\ 0 & 1 & 0 & 10 & 1 & 17 & 120 & 48 & 540 \\ 0 & 0 & 1 & 0 & 2 & 0 & 72 & 48 & 504 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & 0 & 0 & 24 \\ 0 & 0 & 0 & 1 & 1 & -\frac{1}{2} & 16 & 4 & 48 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{3} & \frac{2}{3} & 0\end{pmatrix}\notag\end{equation}$$