Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-74-22-b

·

This is a Fano fourfold of K3 type, appearing as case GM-18 in Bernardara–Fatighenti–Manivel–Tanturri.

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
E^(w4) + E^(2w4) + E^(w1 + w3)
Bundle
O_2(1) + O_2(2) + O_1(1) box U_2^v
ZL62 label
2041.39h
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 22
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 74
$\chi(X, \mathrm{T}_X)$ = -30

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 74
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 8
$\mathrm{c}_4$ = 38
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 30 1 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 + 30 0 x0 + 30 x2
$\mathrm{HH}^{ 3 }$ = x1 + 5 0 0 x1 5
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 12 0 0 x1 - x2 - 11 1 22
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 17t + 38t^{2} + 17t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(662)
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) ^{2} \left( 5t + 1 \right) ^{2} \left( 236800t^{6} + 143488t^{5} + 27968t^{4} + 1960t^{3} + 191t^{2} + 22t - 1 \right) \left( 985388740116480000t^{16} + 2485762895042969600t^{15} + 3022703910100172800t^{14} + 2348523043488661504t^{13} + 1292595692937707520t^{12} + 526558588844282368t^{11} + 161565375530348160t^{10} + 37505478438270784t^{9} + 6568671250929760t^{8} + 861869820094112t^{7} + 84033271413304t^{6} + 6071126227236t^{5} + 329158860972t^{4} + 13684096786t^{3} + 410409567t^{2} + 5751890t - 56795 \right)D^{6} + \left( 4t + 1 \right) \left( 5t + 1 \right) \left( 98002822537024634880000000t^{24} + 334370490082711620812800000t^{23} + 551612914613339551694848000t^{22} + 585214613709353489793024000t^{21} + 446626094381181181100032000t^{20} + 259347902433123364509843456t^{19} + 118149072536817512824963072t^{18} + 42961392816055490576318464t^{17} + 12590945365946705797464064t^{16} + 2990172149811286814212096t^{15} + 577014492294210491724288t^{14} + 90597613375463436713088t^{13} + 11586815638811422424064t^{12} + 1210047088527387617184t^{11} + 103885278606595445824t^{10} + 7451760976613940312t^{9} + 461344441628610360t^{8} + 25819625658738956t^{7} + 1341156238807796t^{6} + 62228134329331t^{5} + 2376495658025t^{4} + 71423607788t^{3} + 1704298580t^{2} + 24313845t - 170385 \right)D^{5} + \left( 16333803756170772480000000000t^{26} + 60900063029940369489920000000t^{25} + 110086398480000227727114240000t^{24} + 128367633868191209491529728000t^{23} + 108117020609085026874687488000t^{22} + 69673716108568096766626365440t^{21} + 35485431153420346803656589312t^{20} + 14558076066105860026909851648t^{19} + 4865951471085816671643500544t^{18} + 1334005407506530196011532288t^{17} + 301088489603606618474928128t^{16} + 56044217525536009741161472t^{15} + 8606854617080950884934272t^{14} + 1090127041012293764028928t^{13} + 113798705740713325818240t^{12} + 9781598061691516149840t^{11} + 690315748429607413440t^{10} + 39561282178911682068t^{9} + 1774778298822139796t^{8} + 55396041885802754t^{7} + 630867300616429t^{6} - 45152285843684t^{5} - 3136252657662t^{4} - 103006711132t^{3} - 2031033073t^{2} - 18749470t + 113590 \right)D^{4} + t \left( 68601975775917244416000000000t^{25} + 248591004601147782594560000000t^{24} + 437656550988031767609344000000t^{23} + 497922885450158535750975488000t^{22} + 409546468345063012655798681600t^{21} + 257674753111483178211229040640t^{20} + 127949824664380782543039889408t^{19} + 51058829879979270187322966016t^{18} + 16551100213488291134064295936t^{17} + 4385991808018413385676652544t^{16} + 953593798380702488504932352t^{15} + 170421730873973288721428480t^{14} + 25057551789079349443056512t^{13} + 3033239574222796376839040t^{12} + 302706732345582418751264t^{11} + 24970025243369731100352t^{10} + 1707445774737580428408t^{9} + 96600552104693516488t^{8} + 4439040637993460012t^{7} + 156670176274335260t^{6} + 3675011192487097t^{5} + 29565510054092t^{4} - 1264574577809t^{3} - 45399724893t^{2} - 479991120t + 968585 \right)D^{3} + t \left( 151577698857264768614400000000t^{25} + 537114470431899728740352000000t^{24} + 926297944427673861520097280000t^{23} + 1033801748400438670314700800000t^{22} + 834654714395453259880438169600t^{21} + 515255700000424133631536005120t^{20} + 250687192494028786850745810944t^{19} + 97809072754520990220542017536t^{18} + 30919315183390663763382894592t^{17} + 7968235446811797713717272576t^{16} + 1680210192897176554202443776t^{15} + 290513915147225733212373504t^{14} + 41248406141371115392320000t^{13} + 4817138562736890608373056t^{12} + 463893447250904029493472t^{11} + 36962194307336664015184t^{10} + 2439935521720505042504t^{9} + 132221263021416305464t^{8} + 5676042949922306080t^{7} + 176034838220217464t^{6} + 3006826065934746t^{5} - 16610398357234t^{4} - 2098441617313t^{3} - 39009559785t^{2} - 180547605t + 283975 \right)D^{2} + 2 t ^{2} \left( 82322370931100693299200000000t^{24} + 286871746232238160740352000000t^{23} + 487161210918783642259947520000t^{22} + 535941868740764013290323968000t^{21} + 426686382280303297516745523200t^{20} + 259618580354806968940243189760t^{19} + 124337951189354832756341735424t^{18} + 47666001672240544183373791232t^{17} + 14773218583965640393138241536t^{16} + 3724214654492621675870396416t^{15} + 766507969541698955603509248t^{14} + 129119283595625158800271872t^{13} + 17838152163527833607962368t^{12} + 2026224740391622581120448t^{11} + 189912459907650242007520t^{10} + 14742140375501427362848t^{9} + 946931313683268247008t^{8} + 49491038913557764152t^{7} + 1996418367585568284t^{6} + 54260763026133048t^{5} + 567253352455254t^{4} - 19955917877160t^{3} - 852368345241t^{2} - 10717809975t - 12800520 \right)D + 192 t ^{3} \left( 350010080489373696000000000t^{23} + 1205798897623535452160000000t^{22} + 2026058223658734379335680000t^{21} + 2206913335584440490393600000t^{20} + 1739984118678715058251366400t^{19} + 1047961923715009667347251200t^{18} + 496293186776612109093175296t^{17} + 187860911126900520686288896t^{16} + 57392940397599905090207744t^{15} + 14236437058687493170764800t^{14} + 2878239592435639921196032t^{13} + 475572613702773007705536t^{12} + 64383581134614234650848t^{11} + 7165029563669777379400t^{10} + 658330666117604452980t^{9} + 50133859844882357712t^{8} + 3155094165052669378t^{7} + 160358521284392644t^{6} + 6159389096866621t^{5} + 149892282308215t^{4} + 746245583890t^{3} - 85283007866t^{2} - 2698460334t - 27602370 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 31.21086 λ ≈ 31.21086; multiplicity 1; Jordan blocks 1 λ ≈ 1.951962 - 9.323017i; multiplicity 1; Jordan blocks 1 λ ≈ 1.951962 + 9.323017i; multiplicity 1; Jordan blocks 1 λ ≈ -5.683524; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 2; Jordan blocks 2 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ -3.715629 - 0.9526273i; multiplicity 1; Jordan blocks 1 λ ≈ -3.715629 + 0.9526273i; multiplicity 1; Jordan blocks 1 2 2
Sector dimension
10
Spectral radius
$31.2108595$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$31.21086$11
$1.951962 - 9.323017\mathrm{i}$11
$1.951962 + 9.323017\mathrm{i}$11
$-5.683524$11
$-5$22
$-4$22
$-3.715629 - 0.9526273\mathrm{i}$11
$-3.715629 + 0.9526273\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{2} \left(\lambda^{6} - 22 \lambda^{5} - 191 \lambda^{4} - 1960 \lambda^{3} - 27968 \lambda^{2} - 143488 \lambda - 236800\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{2} \left(\lambda^{6} - 22 \lambda^{5} - 191 \lambda^{4} - 1960 \lambda^{3} - 27968 \lambda^{2} - 143488 \lambda - 236800\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 52 & 288 & 84 & 252 & 336 & 1856 & 2576 & 9600 \\ 1 & -1 & 6 & 40 & 0 & 44 & 64 & 168 & 204 & 832 \\ 1 & 1 & 4 & 32 & 12 & 32 & 40 & 252 & 360 & 1440 \\ 0 & 1 & 1 & 3 & 0 & 6 & 10 & 20 & 20 & 72 \\ 0 & 1 & 0 & 6 & -1 & 0 & 0 & 44 & 64 & 192 \\ 0 & 0 & 1 & 1 & 1 & -2 & 2 & 8 & 12 & 72 \\ 0 & 0 & 1 & 1 & 0 & 2 & -2 & 16 & 24 & 96 \\ 0 & 0 & 0 & -1 & -2 & 5 & 6 & -11 & -12 & -32 \\ 0 & 0 & 0 & 3 & 2 & -2 & -2 & 12 & 14 & 56 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$