Family 3-241-54-b

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

Fl(2, 3, 7) · 605.5311 (preferred model)

Defining data

Ambient: Fl(2, 3, 7)
Bundle: $(E^{ω_{2}})^{\oplus 2} \oplus E^{ω_{4}} \oplus E^{ω_{6}}$ 2 copies of E^(w2) + E^(w4) + E^(w6)
Bundle: $O (1, 0)^{\oplus 2} \oplus Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor} \oplus ⋀^{3} Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor}$ 2 copies of O(1,0) + Q_1^v ot U_2^v ot wedge^2 U_3^v + wedge^3 Q_1^v ot U_2^v ot wedge^2 U_3^v
ZL62 label: 605.5311

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	54
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	241
$\chi(X, \mathrm{T}_X)$	=	-2

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	241
$\mathrm{c}_1^2\mathrm{c}_2$	=	154
$\mathrm{c}_2^2$	=	104
$\mathrm{c}_1\mathrm{c}_3$	=	50
$\mathrm{c}_4$	=	17

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		9		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 + 40	0	0	x₁ - 14	54
$\mathrm{HH}^{ 4 }$ = 54	0	0	0	0	54
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 49t + 141t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(191)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(267)

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) ^{2} \left( 11t^{3} + 18t^{2} - t + 1 \right) \left( 997t^{5} - 468t^{4} + 361t^{3} + 81t^{2} - 2t - 1 \right) \left( 1589599752363294t^{16} + 1812291462162186t^{15} - 2727585349653885t^{14} - 9217316732195002t^{13} - 8812348311134576t^{12} - 3818318284860838t^{11} - 1556983089141721t^{10} - 1181355892364158t^{9} - 849391717618685t^{8} - 320864621412550t^{7} - 71139338072781t^{6} - 1826982734146t^{5} - 126645509276t^{4} - 73954937114t^{3} - 1720726345t^{2} - 27523834t - 626841 \right)D^{6} + \left( t + 1 \right) \left( 366095950167533151258t^{25} + 1027165636742747121540t^{24} + 118422695042989603569t^{23} - 3376389604852751418714t^{22} - 6565474414094274648626t^{21} - 6020082957781537458968t^{20} - 3732393085946027817252t^{19} - 2513135896777937209973t^{18} - 1835828116182764353372t^{17} - 1088353593199573669763t^{16} - 519308018107157569745t^{15} - 252015515822080646899t^{14} - 122452618090921983740t^{13} - 45338327524681350254t^{12} - 11052095754318215782t^{11} - 1049280978401264466t^{10} - 136949249037818861t^{9} - 25555281818400289t^{8} - 4607518618302462t^{7} - 782205827084197t^{6} - 31758451679924t^{5} - 1208257766264t^{4} - 375327782088t^{3} - 6984112631t^{2} - 78183615t - 1253682 \right)D^{5} + \left( 3050799584729442927150t^{26} + 10386970734608157142170t^{25} + 6012574720203863338563t^{24} - 30454459911790436345558t^{23} - 80989874625664525360697t^{22} - 96335546226139513500824t^{21} - 69804315694541025895994t^{20} - 40719625831946575336583t^{19} - 26956450998745021888568t^{18} - 18957743144049942715340t^{17} - 11015819924609933405105t^{16} - 5069855260077165712386t^{15} - 2060341123278557432877t^{14} - 767804145797437177743t^{13} - 239475993492159440536t^{12} - 46143720276600200112t^{11} - 2535755405240432593t^{10} - 129832709432633093t^{9} - 27684046775954035t^{8} + 2325854935981140t^{7} + 706822919791810t^{6} - 12136755988145t^{5} - 2331126650726t^{4} + 320233663947t^{3} + 6000105407t^{2} + 82008767t + 626841 \right)D^{4} + t \left( 12813358255863660294030t^{25} + 39587586869528471064330t^{24} + 14533409106270465006195t^{23} - 135954518227636175814051t^{22} - 327887846633610701565248t^{21} - 370493818485223335213050t^{20} - 258575844492933759584904t^{19} - 150016869162101648921055t^{18} - 103481443996989927064255t^{17} - 75596010554069717185495t^{16} - 43828091405306096814178t^{15} - 18749443032937283908536t^{14} - 6300009473467669776909t^{13} - 1863294573748161181856t^{12} - 536101986147246635676t^{11} - 98295871698434313948t^{10} - 4463037424687205969t^{9} - 420276830502044816t^{8} - 183351413745786677t^{7} - 10312188915436393t^{6} - 474003094141991t^{5} - 6255312213638t^{4} - 1005689828368t^{3} - 39141388665t^{2} - 716110706t - 215883 \right)D^{3} + 2 t ^{2} \left( 14155710073144615181976t^{24} + 40322048287721560720740t^{23} + 7580029754390226852813t^{22} - 155378603998909241661679t^{21} - 348922588136223976518490t^{20} - 378554134450780653504888t^{19} - 259271620266092541843086t^{18} - 155820044000993633686137t^{17} - 114380933406569579384052t^{16} - 84744621893928823155216t^{15} - 47807206389766023481398t^{14} - 19071920445222729196242t^{13} - 5444232488837745228024t^{12} - 1259718093681769306236t^{11} - 330908815988498362636t^{10} - 58517637515127162090t^{9} - 2533669246887501360t^{8} - 339430033592109358t^{7} - 147871341372732689t^{6} - 5524158847876771t^{5} - 44795304676626t^{4} + 2224184916898t^{3} - 517176011320t^{2} - 19949892589t - 281882276 \right)D^{2} + 8 t ^{2} \left( 3844007476759098088209t^{24} + 10270376543200557084669t^{23} + 452307490600383381132t^{22} - 43014696007655698862052t^{21} - 91757319797644124299462t^{20} - 96515307782769878217897t^{19} - 65582881392233672423741t^{18} - 41200060786160786948819t^{17} - 31803155252917402984910t^{16} - 23563998657594664776082t^{15} - 12916071181214305726129t^{14} - 4879763076565433987166t^{13} - 1228068242613405108608t^{12} - 226241625180656969982t^{11} - 54672719513283189911t^{10} - 9405858178486325682t^{9} - 438202792247311499t^{8} - 71546514078093227t^{7} - 28504109061118527t^{6} - 773853210993746t^{5} + 15793311342686t^{4} + 243989147331t^{3} - 75734800068t^{2} - 2720083155t - 29252580 \right)D + 48 t ^{2} \left( 261497107262523679470t^{24} + 667452252352652391870t^{23} - 40784690869885214946t^{22} - 2958250868394085620126t^{21} - 6095656455469978629236t^{20} - 6273617942435968891560t^{19} - 4252170378633614425915t^{18} - 2772378098118821341313t^{17} - 2212034175495716601574t^{16} - 1631969600760809995306t^{15} - 874960694933996036687t^{14} - 318243417783642225890t^{13} - 73164531403330613590t^{12} - 11134496668227464480t^{11} - 2528358643535054733t^{10} - 425887935604505134t^{9} - 21936921827058286t^{8} - 3864598606310332t^{7} - 1411397771026455t^{6} - 29765350180422t^{5} + 1248263916316t^{4} - 12632574600t^{3} - 3311880736t^{2} - 104965667t - 835788 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 7)

Show multiplicities

Sector dimension: 10
Spectral radius: $9.73438457$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$9.734385$	1	1
$-6.53657 - 2.428077\mathrm{i}$	1	1
$-6.53657 + 2.428077\mathrm{i}$	1	1
$0.7907061 - 4.277176\mathrm{i}$	1	1
$0.7907061 + 4.277176\mathrm{i}$	1	1
$0.669378 - 1.287782\mathrm{i}$	1	1
$0.669378 + 1.287782\mathrm{i}$	1	1
$-1$	2	1, 1
$-0.5814122$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{3} - \lambda^{2} + 18 \lambda + 11\right) \left(\lambda^{5} + 2 \lambda^{4} - 81 \lambda^{3} - 361 \lambda^{2} + 468 \lambda - 997\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{3} - \lambda^{2} + 18 \lambda + 11\right) \left(\lambda^{5} + 2 \lambda^{4} - 81 \lambda^{3} - 361 \lambda^{2} + 468 \lambda - 997\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 6 & 0 & 12 & 6 & 24 & 24 & 60 \\ 1 & -1 & 0 & 4 & 0 & 8 & 4 & 0 & 3 & 32 \\ 1 & 2 & 0 & -2 & 2 & -2 & 2 & 9 & 3 & -16 \\ 0 & 1 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 1 & -2 & -6 & -6 & -12 \\ 0 & 2 & 1 & 0 & 0 & -1 & -1 & 6 & 4 & 0 \\ 0 & 0 & 1 & 1 & 0 & 2 & 1 & 0 & 2 & 12 \\ 0 & 0 & 0 & 2 & -1 & \frac{5}{3} & -\frac{8}{3} & -4 & -2 & \frac{8}{3} \\ 0 & 0 & 0 & 0 & 3 & 3 & 7 & 6 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{9}{4} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta