Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-85-24-b

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 5)
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_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w2 + w4)
Bundle
2 copies of Q_1 box O_2(1)
ZL62 label
2041.mm
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 85
$\chi(X, \mathrm{T}_X)$ = -24

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 85
$\mathrm{c}_1^2\mathrm{c}_2$ = 106
$\mathrm{c}_2^2$ = 131
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 28
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 22 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 + 24 0 x0 + 24 x1
$\mathrm{HH}^{ 3 }$ = x1 + 21 0 0 x1 + 15 6
$\mathrm{HH}^{ 4 }$ = 24 0 0 0 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 45t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(589)
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( 2t + 1 \right) \left( 3t + 1 \right) \left( 609100t^{8} + 1047290t^{7} + 715429t^{6} + 244493t^{5} + 42878t^{4} + 3602t^{3} + 169t^{2} + 9t - 1 \right) \left( 625309679414016t^{17} + 2407124555132160t^{16} + 4296379206945024t^{15} + 4611813546629760t^{14} + 3299700248956216t^{13} + 1669826813354400t^{12} + 630293066096776t^{11} + 191981631227092t^{10} + 52672350001614t^{9} + 13707223672632t^{8} + 3105058189484t^{7} + 536969395825t^{6} + 64532190977t^{5} + 5053967477t^{4} + 241916264t^{3} + 6121885t^{2} + 30301t + 136 \right)D^{6} + \left( 47990391842115720345600t^{27} + 292707992032890871265280t^{26} + 847768231556004330888192t^{25} + 1541994357363089492865792t^{24} + 1968820495631554521854400t^{23} + 1873090988828427779076672t^{22} + 1378935753680774643719008t^{21} + 807771123409511633852312t^{20} + 386101025677338789985088t^{19} + 154785887352323720556400t^{18} + 53748222827010451258472t^{17} + 16688045313481520107182t^{16} + 4713157281896738456628t^{15} + 1199685376532358426310t^{14} + 266953077506797035460t^{13} + 50098295510761135087t^{12} + 7687706935281706420t^{11} + 941403427887591315t^{10} + 90215691582485178t^{9} + 6672794639982410t^{8} + 382473190068492t^{7} + 18047437246015t^{6} + 808246435125t^{5} + 35794872743t^{4} + 1247215720t^{3} + 24778472t^{2} + 91991t + 272 \right)D^{5} + \left( 399919932017631002880000t^{27} + 2342316074306626150041600t^{26} + 6513007620518804543843328t^{25} + 11354440817027748359457024t^{24} + 13865554853969983837306176t^{23} + 12592357495009060950545600t^{22} + 8841695217196818616353056t^{21} + 4947435159258872041198216t^{20} + 2272350481960778855846496t^{19} + 886326478112682344844416t^{18} + 304854425614165041639924t^{17} + 95196158978572081991242t^{16} + 27071520631196003713856t^{15} + 6834310201671692920434t^{14} + 1474850259726326849911t^{13} + 262853667439111501965t^{12} + 37669538921068960624t^{11} + 4247604706098264971t^{10} + 368733022893260711t^{9} + 23916709261226770t^{8} + 1093095559166262t^{7} + 29631558165354t^{6} + 64130467619t^{5} - 29361086484t^{4} - 1151814226t^{3} - 21178127t^{2} - 68106t - 136 \right)D^{4} + t \left( 1679663714474050212096000t^{26} + 9517900776395197165516800t^{25} + 25599321175000456445875200t^{24} + 43103676520994830068783360t^{23} + 50749250273547965477825856t^{22} + 44380629390405296831545536t^{21} + 30008192722044762324648736t^{20} + 16214249290979328084529032t^{19} + 7244457498721680494043264t^{18} + 2784636562913415889468816t^{17} + 957933972634487201834184t^{16} + 301238901946555724849066t^{15} + 85600583780736449086596t^{14} + 21216557243750306970858t^{13} + 4413706252850037382148t^{12} + 747575670214041596529t^{11} + 100806205331916175930t^{10} + 10622424889617320977t^{9} + 857864594455887466t^{8} + 51732104091699630t^{7} + 2233422728354808t^{6} + 64064289421575t^{5} + 1073054273314t^{4} + 9198654507t^{3} + 55428734t^{2} + 546414t + 432 \right)D^{3} + 2 t ^{2} \left( 1855628484561807853363200t^{25} + 10244684450182111640110080t^{24} + 26838041279526849648027648t^{23} + 43955337805967555871499776t^{22} + 50266423364267938566057216t^{21} + 42659843729694271098596224t^{20} + 28006662753303948334129984t^{19} + 14739741077670922361899408t^{18} + 6462347801362704847630608t^{17} + 2465708728096561540897784t^{16} + 850711904373831435350016t^{15} + 268483104876905363230988t^{14} + 75741160465526050402604t^{13} + 18350453397223864903142t^{12} + 3680772434449173473082t^{11} + 595080866576505451697t^{10} + 76043250262174917962t^{9} + 7550251275847833967t^{8} + 571098067374216974t^{7} + 31955861425170080t^{6} + 1256825383497200t^{5} + 31729452965262t^{4} + 457201084748t^{3} + 4719111169t^{2} + 57208332t + 266033 \right)D^{2} + 4 t ^{2} \left( 1007798228684430127257600t^{25} + 5456332835172524134241280t^{24} + 14012662746368830329971712t^{23} + 22471366668588656327753472t^{22} + 25132072588104654649344192t^{21} + 20846144653781562471877824t^{20} + 13383981524756142842952736t^{19} + 6909187129128900286624456t^{18} + 2990809480766999718129904t^{17} + 1137154213856746582125520t^{16} + 393529931761240692276088t^{15} + 124278874045141786265038t^{14} + 34722265176406192906632t^{13} + 8230903587587657652298t^{12} + 1599351501024060504904t^{11} + 248687015482064969542t^{10} + 30401953925730654580t^{9} + 2874728123122348202t^{8} + 205983099787514404t^{7} + 10818497954817801t^{6} + 391529553190960t^{5} + 8694432609544t^{4} + 101904567860t^{3} + 1052592199t^{2} + 14679872t + 57664 \right)D + 8 t ^{2} \left( 205673107894781658624000t^{25} + 1098703049121821592115200t^{24} + 2783019069407069534767104t^{23} + 4397609012872772799747072t^{22} + 4841660389535988715034496t^{21} + 3951357154623183531477888t^{20} + 2497247181032563559776320t^{19} + 1271986141605353916206928t^{18} + 546055698458883849439392t^{17} + 207297768979865061587112t^{16} + 71905176217889000841144t^{15} + 22690736991827636245392t^{14} + 6282805826635124157760t^{13} + 1463031112155477495426t^{12} + 277287871877726211460t^{11} + 41834230371956564884t^{10} + 4941844655026817476t^{9} + 449846659224472616t^{8} + 30886165945519458t^{7} + 1541904939977129t^{6} + 52142308804986t^{5} + 1039692182058t^{4} + 9989754714t^{3} + 93718135t^{2} + 1217584t + 4352 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 24.98666 λ ≈ 24.98666; multiplicity 1; Jordan blocks 1 λ ≈ -0.821904 - 11.11718i; multiplicity 1; Jordan blocks 1 λ ≈ -0.821904 + 11.11718i; multiplicity 1; Jordan blocks 1 λ ≈ -3.302489 - 0.8314822i; multiplicity 1; Jordan blocks 1 λ ≈ -3.302489 + 0.8314822i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.832765 - 0.3704973i; multiplicity 1; Jordan blocks 1 λ ≈ -2.832765 + 0.3704973i; multiplicity 1; Jordan blocks 1 λ ≈ -2.072339; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$24.9866569$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$24.98666$11
$-0.821904 - 11.11718\mathrm{i}$11
$-0.821904 + 11.11718\mathrm{i}$11
$-3.302489 - 0.8314822\mathrm{i}$11
$-3.302489 + 0.8314822\mathrm{i}$11
$-3$11
$-2.832765 - 0.3704973\mathrm{i}$11
$-2.832765 + 0.3704973\mathrm{i}$11
$-2.072339$11
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{8} - 9 \lambda^{7} - 169 \lambda^{6} - 3602 \lambda^{5} - 42878 \lambda^{4} - 244493 \lambda^{3} - 715429 \lambda^{2} - 1047290 \lambda - 609100\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{8} - 9 \lambda^{7} - 169 \lambda^{6} - 3602 \lambda^{5} - 42878 \lambda^{4} - 244493 \lambda^{3} - 715429 \lambda^{2} - 1047290 \lambda - 609100\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 26 & 174 & 87 & 135 & 216 & 928 & 960 & 3820 \\ 1 & 0 & 6 & 8 & -6 & 18 & 24 & 30 & 30 & 320 \\ 1 & 0 & 2 & 22 & 16 & 18 & 30 & 168 & 195 & 768 \\ 0 & 1 & 1 & 2 & -2 & 4 & 6 & 0 & -10 & 0 \\ 0 & 1 & 0 & 6 & 1 & 3 & 6 & 24 & 20 & 72 \\ 0 & 0 & 1 & 0 & 1 & -3 & 0 & 8 & 10 & 48 \\ 0 & 0 & 1 & 0 & 1 & 2 & 0 & 12 & 20 & 84 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & 3 & \frac{9}{2} & 0 & 0 & 12 \\ 0 & 0 & 0 & 1 & 1 & -\frac{2}{5} & -\frac{3}{5} & 4 & 2 & \frac{56}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{5}{4} & 0\end{pmatrix}\notag\end{equation}$$