Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-134-34-a

·

This is a Fano fourfold of K3 type, appearing as case K3-57 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × Fl(1, 3, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 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^(w6) + E^(2w4) + E^(w1 + 2w2)
Bundle
2 copies of wedge^2 Q_2,1^v ot wedge^2 U_2,2^v ot U_2,3^v + O_2(0,2) + O_1(1) box O_2(2,0)
ZL62 label
154.559x301
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;3,5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;2,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 34
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 134
$\chi(X, \mathrm{T}_X)$ = -18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 134
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 114
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = 36
Hodge diamond
1
00
0 4 0
0 0 0 0
0 1 24 1 0
0 0 0 0
0 4 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 + 12 0 0 x1 12
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 29 0 0 x1 - x2 - 5 0 34
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 29t + 74t^{2} + 29t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(592)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(615)

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) ^{4} \left( 25t^{2} + 5t - 1 \right) \left( 63t^{2} + 15t + 1 \right) \left( 269t^{3} - 92t^{2} - 14t + 1 \right) \left( 26740364824177278075t^{16} + 40516273852085135664t^{15} + 63834940390439684628t^{14} + 63455439791202671061t^{13} + 32029083028997169741t^{12} + 7895693060801939040t^{11} + 602606280358765757t^{10} - 133832585972901237t^{9} - 33784180867172886t^{8} - 2354861520803685t^{7} + 36958810537685t^{6} + 13345237823050t^{5} + 720472032124t^{4} + 22762620851t^{3} + 297484140t^{2} + 1899832t - 4288 \right)D^{8} + \left( 3t + 1 \right) ^{3} \left( 1223556199223397295149967500t^{24} + 2229954792012290148528922200t^{23} + 3466768333034040934363581420t^{22} + 3748399668705296743549846605t^{21} + 2107479424633111635193836333t^{20} + 422661961358591031078504543t^{19} - 141109455465443341748572452t^{18} - 104797100382197148137263689t^{17} - 24303605996087117018391729t^{16} - 1431757267633381856665825t^{15} + 454316895158194225070978t^{14} + 102421181082967291551436t^{13} + 6802667537224207296350t^{12} - 212576037182770144694t^{11} - 52970040072646240112t^{10} - 3555247650129917679t^{9} - 290102531908239943t^{8} - 15733558825244003t^{7} + 931078976673354t^{6} + 151154345963015t^{5} + 6873817689237t^{4} + 198122861539t^{3} + 2313234200t^{2} + 13427464t - 25728 \right)D^{7} + \left( 3t + 1 \right) ^{2} \left( 55671807064664576929323521250t^{25} + 114269652141737899352424127200t^{24} + 186832772572306118331036487680t^{23} + 219021111603752632835674950711t^{22} + 148757107227509266478985913362t^{21} + 52542409838692613839941002202t^{20} + 4998169632644367020443357749t^{19} - 3227381348379473018265027183t^{18} - 1378093458750510752665719048t^{17} - 202007015937471093598538076t^{16} + 5027875792728868806485659t^{15} + 6205931442504939393156466t^{14} + 908041070809113184923272t^{13} + 45618145200313961947120t^{12} - 2491188216840707555742t^{11} - 470039902202601782845t^{10} - 18432427939783133678t^{9} + 1174799329516719820t^{8} + 136113670611057217t^{7} + 1118220353361775t^{6} - 409509692807890t^{5} - 20734598750646t^{4} - 592115832511t^{3} - 6356933296t^{2} - 32158280t + 55744 \right)D^{6} + \left( 3t + 1 \right) \left( 1387512729919332532700063145000t^{26} + 3190297020949953989331551809200t^{25} + 5465754921062946998225818141740t^{24} + 6858192316641024634589262436767t^{23} + 5348455234502264565188723723097t^{22} + 2465909624261536231625710861014t^{21} + 599089640564646396913370711661t^{20} + 20238073435274157057742916670t^{19} - 31969773312676960313973274713t^{18} - 9155311521067209455377531884t^{17} - 773593143808827198494488419t^{16} + 128169752945697653759249801t^{15} + 41214786985422949328562356t^{14} + 4525746340793808490978558t^{13} + 169884468340787169368150t^{12} - 13143668141935131649151t^{11} - 2208484165123463096391t^{10} - 176336749903059391888t^{9} - 11491997209105004325t^{8} - 526795701941314404t^{7} - 8441453856951197t^{6} + 457960284334936t^{5} + 26082354586089t^{4} + 727840732933t^{3} + 7187579384t^{2} + 31666648t - 51456 \right)D^{5} + \left( 20600709837274534409116215305625t^{27} + 52756319594607457315671022787550t^{26} + 94713999644301916434753670807635t^{25} + 126125477545383456708344596906923t^{24} + 109857340625584788336608014928211t^{23} + 60504830371788072042292740136809t^{22} + 20398811155090294693003131005721t^{21} + 3543985439164919501422513320213t^{20} - 75515272837206152891708322522t^{19} - 185099875831469381839194201633t^{18} - 38014735300485741404917662816t^{17} - 1638713281487126561800391796t^{16} + 675998145578703953464774963t^{15} + 145633366397245696783037536t^{14} + 12251513621799910078237661t^{13} + 147700294039064839357531t^{12} - 69513301944034910066073t^{11} - 7693990554425223850029t^{10} - 426446082616115691343t^{9} - 12338160338884079175t^{8} - 55762687271991334t^{7} + 3786295920364995t^{6} - 115774601475762t^{5} - 9304694160936t^{4} - 295803304078t^{3} - 2867390932t^{2} - 11009120t + 17152 \right)D^{4} + 4 t \left( 15436079120352574426288202488125t^{26} + 38628377541340285225711551565275t^{25} + 69604439453450070791401036218465t^{24} + 92697819777396541384296568144470t^{23} + 79892852261198159602208633062941t^{22} + 43395588169892830777295248566462t^{21} + 14518435418679578560560525701628t^{20} + 2597591870017580233440384007761t^{19} + 19862163523589304768989095890t^{18} - 104623753503575863461726720342t^{17} - 22945661583915696052531003677t^{16} - 1425858419153935671052696950t^{15} + 255640670010230063490985403t^{14} + 61523817891927156314125475t^{13} + 5040275533960137910876762t^{12} + 43203679733077105866743t^{11} - 28444117598686847458649t^{10} - 2971113860839622351371t^{9} - 167853885146096255774t^{8} - 6079783690722169922t^{7} - 135769329958049914t^{6} - 2079406785163423t^{5} - 23403540585000t^{4} - 306485128914t^{3} - 2835646520t^{2} + 200192t - 2432 \right)D^{3} + 12 t ^{2} \left( 9033209529816536380768422560625t^{25} + 22185840799726041996564913058850t^{24} + 40095700847192078502563011458915t^{23} + 53363676785340393615544019784693t^{22} + 45530683810221926418611946093333t^{21} + 24374185320554733735234050808087t^{20} + 8036587657433183128525557232893t^{19} + 1428572873822491430623521379197t^{18} + 19327815626341450620776255772t^{17} - 53156452434603314307124780329t^{16} - 12050281309745426576772780462t^{15} - 937376862813520469873983260t^{14} + 70416059555117008800741177t^{13} + 22127419926310921605593904t^{12} + 1861240135913958291414405t^{11} + 20835263824517086504789t^{10} - 9231199044335181529139t^{9} - 958021258349791530355t^{8} - 53458759717437498719t^{7} - 1903937854615879471t^{6} - 42261302861412572t^{5} - 632362307078889t^{4} - 6442752229812t^{3} - 76086731088t^{2} - 643743744t - 102912 \right)D^{2} + 144 t ^{3} \left( 698344700706754006206843950625t^{24} + 1690359518693768730099855880725t^{23} + 3062007116972039642982337369845t^{22} + 4071566053377091584280167412446t^{21} + 3443754048962863811822945971668t^{20} + 1819272390278283192982338950004t^{19} + 590226528438925757403205666851t^{18} + 102741878561462223308941584906t^{17} + 1032713394786192664446019752t^{16} - 3865421203710191855518195515t^{15} - 879262000018335067855661673t^{14} - 76551960102966670237838016t^{13} + 2282672100435753506142099t^{12} + 1170625662114998701681383t^{11} + 104511675008549217361620t^{10} + 1886224419282287477413t^{9} - 399642849573058710290t^{8} - 43586898357972186919t^{7} - 2442646285931599481t^{6} - 85463644264578427t^{5} - 1880602442582072t^{4} - 27266237189076t^{3} - 250915459320t^{2} - 2218037364t - 15491472 \right)D + 1728 t ^{3} \left( 21412233486409452665124431250t^{24} + 51287824297844273520157604625t^{23} + 93060896799951433800209751300t^{22} + 123640658706652765631987642865t^{21} + 103875239464245202953303823815t^{20} + 54298004037394412374398117072t^{19} + 17368289940937667090972082147t^{18} + 2953819264780094546892051774t^{17} + 10863213204736110241722210t^{16} - 115705205856360479117493186t^{15} - 26064680934838912420518606t^{14} - 2392359967104831077382390t^{13} + 21176561755618048096310t^{12} + 27448639880617925128523t^{11} + 2614262852514578061820t^{10} + 66690606114982273253t^{9} - 7069967817659689787t^{8} - 859637411334768280t^{7} - 49226868521889339t^{6} - 1701750091256376t^{5} - 37331265293016t^{4} - 538753484016t^{3} - 4976071488t^{2} - 34006464t - 154368 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Fl(1, 3, 6)

|λ| = 18.23606 λ ≈ 18.23606; multiplicity 1; Jordan blocks 1 λ ≈ 8.09017; multiplicity 1; Jordan blocks 1 λ ≈ -7.5 - 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -7.5 + 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -6.504034; multiplicity 1; Jordan blocks 1 λ ≈ -3.09017; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 5; Jordan blocks 1, 1, 1, 2 λ ≈ 2.267976; multiplicity 1; Jordan blocks 1 5
Sector dimension
12
Spectral radius
$18.2360583$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$18.23606$11
$8.09017$11
$-7.5 - 2.598076\mathrm{i}$11
$-7.5 + 2.598076\mathrm{i}$11
$-6.504034$11
$-3.09017$11
$-3$51, 1, 1, 2
$2.267976$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{5} \left(\lambda^{2} + 15 \lambda + 63\right) \left(\lambda^{2} - 5 \lambda - 25\right) \left(\lambda^{3} - 14 \lambda^{2} - 92 \lambda + 269\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda^{2} + 15 \lambda + 63\right) \left(\lambda^{2} - 5 \lambda - 25\right) \left(\lambda^{3} - 14 \lambda^{2} - 92 \lambda + 269\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 20 & 8 & 54 & 24 & 48 & 0 & 144 & 0 & 32 & 160 \\ 1 & -1 & 3 & 0 & 16 & 8 & 12 & 0 & 36 & 0 & 0 & 32 \\ 1 & 2 & 0 & 2 & 8 & 0 & 8 & -4 & 24 & 0 & 0 & 0 \\ 1 & 0 & 2 & -2 & 8 & 8 & 20 & 12 & 54 & 24 & 48 & 144 \\ 0 & 1 & \frac{3}{2} & 0 & 0 & 2 & 1 & 0 & 8 & -4 & -2 & 0 \\ 0 & 1 & -\frac{1}{2} & 1 & 1 & -3 & 2 & 0 & 16 & 12 & 14 & 36 \\ 0 & 0 & 2 & 1 & 2 & 2 & 0 & 2 & 8 & 0 & 8 & 24 \\ 0 & 0 & 0 & 1 & 0 & 0 & 2 & -2 & 8 & 8 & 12 & 30 \\ 0 & 0 & 0 & 0 & 2 & 1 & \frac{3}{2} & 0 & 0 & 2 & 1 & 8 \\ 0 & 0 & 0 & 0 & 0 & 1 & -\frac{1}{2} & 1 & 1 & -3 & 2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 2 & 2 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$