Family 3-274-60-a

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

Gr(2, 5) × Gr(2, 6) · 4151.wLw3 (preferred model)

Defining data

Ambient: Gr(2, 5) × Gr(2, 6)
Bundle: $E^{ω_{1} + ω_{5}} \oplus E^{ω_{4} + ω_{5}}$ E^(w1 + w5) + E^(w4 + w5)
Bundle: $U_{1}^{\lor} ⊠ U_{2}^{\lor} \oplus Q_{1} ⊠ U_{2}^{\lor}$ U_1^v box U_2^v + Q_1 box U_2^v
ZL62 label: 4151.wLw3

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;6) \\ \det\mathcal{E} &\cong \mathcal{O}(4;5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	60
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	274
$\chi(X, \mathrm{T}_X)$	=	2

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	274
$\mathrm{c}_1^2\mathrm{c}_2$	=	160
$\mathrm{c}_2^2$	=	106
$\mathrm{c}_1\mathrm{c}_3$	=	52
$\mathrm{c}_4$	=	16

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		8		0		0
	0		0		0		0
		0		3		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ - 2	0	x₀ - 2	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 43	0	0	x₁ - 25	68
$\mathrm{HH}^{ 4 }$ = 60	0	0	0	0	60
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$

$\displaystyle\frac{ 1 + 55t + 162t^{2} + 55t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(158)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(220)

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( t - 1 \right) \left( 11t^{3} - 18t^{2} - t - 1 \right) \left( 19t^{3} - 22t^{2} + 11t - 1 \right) \left( 35t^{3} - 38t^{2} - 13t - 1 \right) \left( 29855618820t^{17} + 195339223496t^{16} - 27007723625388t^{15} + 124890955854636t^{14} - 245263179755319t^{13} + 243363537301532t^{12} - 139663955048745t^{11} + 70258666616563t^{10} - 40914061886230t^{9} + 15611605197858t^{8} - 2874558923213t^{7} + 973322633748t^{6} - 201050392188t^{5} + 39100736125t^{4} + 1154836416t^{3} + 82233895t^{2} + 676599t + 18221 \right)D^{6} + \left( 4586270885034300t^{27} + 13318088811913100t^{26} - 4644089508837147456t^{25} + 40407244472185883092t^{24} - 160571429872047970665t^{23} + 377163690753051577207t^{22} - 578433670141408842530t^{21} + 614371719290118201419t^{20} - 480776513844597938687t^{19} + 303799167069334389045t^{18} - 170200423407425095639t^{17} + 84304361060375732033t^{16} - 33597204471277072587t^{15} + 10284138395125369749t^{14} - 2684617033421914895t^{13} + 688177826430005665t^{12} - 149425021159207550t^{11} + 19344792701010406t^{10} - 606261183615443t^{9} - 138419728919500t^{8} - 19903483857953t^{7} - 13699260134915t^{6} + 898230598530t^{5} - 259849483129t^{4} - 6030439760t^{3} - 334797898t^{2} - 1956913t - 36442 \right)D^{5} + \left( 38218924041952500t^{27} + 148930276898898500t^{26} - 41268518649633649744t^{25} + 338724409380854055848t^{24} - 1272001336978690687795t^{23} + 2816295982866745983965t^{22} - 4060333759550627884630t^{21} + 4068982644990295356348t^{20} - 3065906908548228123998t^{19} + 1939381463701190522156t^{18} - 1110683675837640046543t^{17} + 547795115946825883621t^{16} - 212042299648082616966t^{15} + 65313237756089248244t^{14} - 18414215656173201291t^{13} + 4867454799862365969t^{12} - 996933810408758427t^{11} + 124825024202294931t^{10} - 4459298474127643t^{9} + 277141326544279t^{8} - 236573028516151t^{7} + 9643501751725t^{6} - 2868986271608t^{5} + 302251422762t^{4} + 5570015143t^{3} + 329436393t^{2} + 1391103t + 18221 \right)D^{4} + t \left( 160519480976200500t^{26} + 750729634414128500t^{25} - 181724237000153920000t^{24} + 1421880889653135138396t^{23} - 5096407128859772385291t^{22} + 10733710996121919825893t^{21} - 14674693285910022520210t^{20} + 13999053558575105526657t^{19} - 10242777657180499920197t^{18} + 6489068452134388388587t^{17} - 3737311344051595545601t^{16} + 1803805608043774374095t^{15} - 680188238578427604397t^{14} + 216007318585092660541t^{13} - 65953693137860673839t^{12} + 17854104456067454213t^{11} - 3500662361154422802t^{10} + 390680242192426880t^{9} - 5699853827999585t^{8} - 763672306737410t^{7} - 305989594597061t^{6} - 4028208066809t^{5} - 679757758232t^{4} - 33323637745t^{3} - 688164750t^{2} - 18525822t - 1463 \right)D^{3} + 2 t ^{2} \left( 177335807554659600t^{25} + 935220304116460900t^{24} - 207811162289073010624t^{23} + 1565654643202825437424t^{22} - 5407023709105744734760t^{21} + 10928628880508650197008t^{20} - 14290079793952607059713t^{19} + 13086646464095987367297t^{18} - 9361582716521474040795t^{17} + 5931878986311335928886t^{16} - 3395891229060668723889t^{15} + 1589961986999270679508t^{14} - 587074184127033238419t^{13} + 194857922165935764949t^{12} - 63704410699818862574t^{11} + 17526560577574435529t^{10} - 3358830211168258099t^{9} + 345775210128121307t^{8} - 3792146875470402t^{7} - 468777284540598t^{6} - 110920227470585t^{5} - 4707798838270t^{4} - 282136371392t^{3} - 12749471112t^{2} - 297566134t - 7416906 \right)D^{2} + 4 t ^{2} \left( 96311688585720300t^{25} + 550046564747486600t^{24} - 115655299011998890016t^{23} + 846998823868367513936t^{22} - 2843806022726213060565t^{21} + 5565135975736336863063t^{20} - 7021898510803877992881t^{19} + 6225049277631033053546t^{18} - 4378538725126052294914t^{17} + 2771229667329761648419t^{16} - 1567629541118745233983t^{15} + 711133187230903815378t^{14} - 259061433202968339068t^{13} + 90281370903175381737t^{12} - 31031461208970065189t^{11} + 8576664668281824738t^{10} - 1607195381349999391t^{9} + 154219247598160872t^{8} - 1612513566539858t^{7} - 39058918656996t^{6} - 15595275035799t^{5} - 2952939246911t^{4} - 77812020097t^{3} - 3474920372t^{2} - 69635125t - 1603448 \right)D + 8 t ^{2} \left( 19655446650147000t^{25} + 118062493336164000t^{24} - 23987011748801500992t^{23} + 172295121286548833916t^{22} - 567266010596204576334t^{21} + 1084708551377306881338t^{20} - 1333371075857815502162t^{19} + 1154192282725672780938t^{18} - 802196930373582639304t^{17} + 506866338396030839938t^{16} - 283112423184665532573t^{15} + 124960929553155809883t^{14} - 45204764402522887883t^{13} + 16461014705578248860t^{12} - 5848566764024240723t^{11} + 1613764692859833464t^{10} - 296020451782354379t^{9} + 26756275571801441t^{8} - 287380160805506t^{7} + 20762797948087t^{6} + 50706273275t^{5} - 595138619963t^{4} - 10704742064t^{3} - 488694862t^{2} - 7057379t - 145768 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 5) × Gr(2, 6)

Sector dimension: 10
Spectral radius: $8.72903154$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$8.729032$	1	1
$-6.864516 - 0.9419768\mathrm{i}$	1	1
$-6.864516 + 0.9419768\mathrm{i}$	1	1
$-0.7907061 - 4.277176\mathrm{i}$	1	1
$-0.7907061 + 4.277176\mathrm{i}$	1	1
$1.135484 - 0.9419768\mathrm{i}$	1	1
$1.135484 + 0.9419768\mathrm{i}$	1	1
$1$	1	1
$0.7290315$	1	1
$0.5814122$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda - 1\right) \left(\lambda^{3} + \lambda^{2} + 18 \lambda - 11\right) \left(\lambda^{3} + 13 \lambda^{2} + 38 \lambda - 35\right) \left(\lambda^{3} - 11 \lambda^{2} + 22 \lambda - 19\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 0 & 6 & 3 & 6 & 32 & 24 & 0 \\ 1 & -1 & 0 & 4 & 4 & 4 & 0 & 12 & 6 & 0 \\ 1 & 1 & 0 & 0 & 4 & 0 & 4 & -6 & -3 & 16 \\ 0 & 1 & -1 & -1 & -1 & -1 & 2 & 4 & 4 & 6 \\ 0 & 1 & 2 & 0 & 0 & 0 & -2 & 0 & 0 & -6 \\ 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 12 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 4 & 2 & 0 \\ 0 & 0 & 0 & 1 & 10 & 3 & 8 & 3 & 3 & 8 \\ 0 & 0 & 0 & 1 & -7 & -1 & -8 & -4 & -4 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{2} & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta