Family 3-166-40-a

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

ℙ²P2 × Fl(1, 4, 5) · 2048.rol (preferred model)

Defining data

Ambient: ℙ²P2 × Fl(1, 4, 5)
Bundle: $E^{ω_{3} + ω_{6}} \oplus E^{ω_{2} + ω_{3}} \oplus E^{ω_{2} + ω_{6}}$ E^(w3 + w6) + E^(w2 + w3) + E^(w2 + w6)
Bundle: $O_{2} (1, 1) \oplus Q_{1} ⊠ O_{2} (1, 0) \oplus Q_{1} ⊠ O_{2} (0, 1)$ O_2(1,1) + Q_1 box O_2(1,0) + Q_1 box O_2(0,1)
ZL62 label: 2048.rol

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3,3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	40
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	166
$\chi(X, \mathrm{T}_X)$	=	-12

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	166
$\mathrm{c}_1^2\mathrm{c}_2$	=	136
$\mathrm{c}_2^2$	=	108
$\mathrm{c}_1\mathrm{c}_3$	=	40
$\mathrm{c}_4$	=	22

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		14		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 + 12	0	x₀ + 12	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 31	0	0	x₁ + 5	26
$\mathrm{HH}^{ 4 }$ = 40	0	0	0	0	40
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 35t + 94t^{2} + 35t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(348)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(425)

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( 2t + 1 \right) ^{2} \left( 648t^{3} - 108t^{2} + 18t - 1 \right) \left( 61t^{4} + 93t^{3} + 53t^{2} + 11t + 1 \right) \left( 13561960320888336t^{17} + 33409327626832608t^{16} + 53265260674042488t^{15} + 66288359313341676t^{14} + 57830352821884224t^{13} + 36388312168548726t^{12} + 16391900408051742t^{11} + 4651205563077486t^{10} + 920067127968417t^{9} + 413453748140220t^{8} + 223336545795327t^{7} + 61993623646357t^{6} + 8337663922770t^{5} + 517889701229t^{4} + 15723213720t^{3} + 621648717t^{2} - 3067544t + 9296 \right)D^{6} + \left( 2t + 1 \right) \left( 22515241037691114107136t^{26} + 108052559176838894297856t^{25} + 266698446189773267803392t^{24} + 463414217350781519490816t^{23} + 619995702101422522582512t^{22} + 647801848378411281993312t^{21} + 529611343649197324390872t^{20} + 338758092531469274229792t^{19} + 168254069721288192920736t^{18} + 64823049055934127030492t^{17} + 20844222201780360650886t^{16} + 7084647926173800976836t^{15} + 2882569372306876156815t^{14} + 1085984112658939035438t^{13} + 304767893952132134081t^{12} + 59628160816846113208t^{11} + 8077803008281059072t^{10} + 772159714529246489t^{9} + 55369020798497820t^{8} + 4059904789800195t^{7} + 496820933385504t^{6} + 55746158171214t^{5} + 3074909424174t^{4} + 72681456784t^{3} + 2502434708t^{2} - 9202632t + 18592 \right)D^{5} + \left( 375254017294851901785600t^{27} + 1831707779813003172476160t^{26} + 4650334333422192966204864t^{25} + 8366094970128323466426432t^{24} + 11628396930943926820412928t^{23} + 12686529226697602859280240t^{22} + 10916853060794610951460944t^{21} + 7423990252269366894942240t^{20} + 3969515030488471368204732t^{19} + 1670977024597987668670116t^{18} + 588748123888153048444710t^{17} + 209992143856852694677626t^{16} + 86973985828705352925996t^{15} + 34951584668444883757479t^{14} + 11031029089403679198844t^{13} + 2508779282807886649489t^{12} + 399202185918226044877t^{11} + 43723887985445261252t^{10} + 3266095266647868136t^{9} + 162409993308574606t^{8} + 2461793091908232t^{7} - 711006169335817t^{6} - 88488170143346t^{5} - 4197505417957t^{4} - 80669650010t^{3} - 2124507855t^{2} + 7183408t - 9296 \right)D^{4} + t \left( 1576066872638377987499520t^{26} + 7175748465334878969242880t^{25} + 17423692889814719785319040t^{24} + 30576909925933863298005888t^{23} + 41566402314915313701922848t^{22} + 44179272919292670665505072t^{21} + 36958052309116429454717520t^{20} + 24365260177143621728430168t^{19} + 12562954300551488010958512t^{18} + 5122461473190462434667552t^{17} + 1835414538413396169323820t^{16} + 722644755263970345375906t^{15} + 320500233292568538198678t^{14} + 125530838111719579033773t^{13} + 36869996779242988439360t^{12} + 7687298246444977424575t^{11} + 1113670053824321453324t^{10} + 110282988293271701926t^{9} + 7417651056476721437t^{8} + 354549680289183322t^{7} + 13510986653851021t^{6} + 406171259356808t^{5} + 8667646306210t^{4} - 127172278096t^{3} - 6035831204t^{2} + 19330136t + 11424 \right)D^{3} + 2 t ^{2} \left( 1741178640248112824285184t^{25} + 7490146889066852727733248t^{24} + 17582862341798219105264160t^{23} + 30298234911233930812845216t^{22} + 40440654399780498572917776t^{21} + 42083572679249963884220736t^{20} + 34450782340153952421321696t^{19} + 22166550666571377994025616t^{18} + 11101418082605916650294352t^{17} + 4436893185857743405112628t^{16} + 1643568050923153390173396t^{15} + 702853643059362419371770t^{14} + 319638157735261783763898t^{13} + 120134939802145121343279t^{12} + 32995459240950062335206t^{11} + 6379854797616911752538t^{10} + 854074129441516330516t^{9} + 78157416125201870468t^{8} + 4923168164418322727t^{7} + 229772835940296320t^{6} + 8540756303228774t^{5} + 193996597447568t^{4} + 1739948943608t^{3} - 102605660026t^{2} - 674782578t + 5020504 \right)D^{2} + 8 t ^{2} \left( 472820061791513396249856t^{25} + 1946930819761932981335424t^{24} + 4459003416048454256556720t^{23} + 7583212080634320139319568t^{22} + 9976888993243918801264632t^{21} + 10215912743502777350930664t^{20} + 8230725767476301915674008t^{19} + 5197900842337247160164604t^{18} + 2544679205768592683849268t^{17} + 1005472434581685238477896t^{16} + 386467595485858778161104t^{15} + 175271502149707070737155t^{14} + 79930947790803081615498t^{13} + 28847252015598160493580t^{12} + 7490651302914425707134t^{11} + 1362112780745993162783t^{10} + 171199807098800683851t^{9} + 14772818878143574779t^{8} + 898772085068196514t^{7} + 42384053667260902t^{6} + 1521534928941081t^{5} + 22250047839208t^{4} - 351839710577t^{3} - 20692385898t^{2} + 5157776t + 464800 \right)D + 48 t ^{2} \left( 32164630053844448724480t^{25} + 128444395516583765310720t^{24} + 289272271699540475521632t^{23} + 487570191839024693015232t^{22} + 634828991241729175487424t^{21} + 642675100768280878780224t^{20} + 512109167469133008406248t^{19} + 319086321129746261510328t^{18} + 153637979627567241916236t^{17} + 60343421173852918941126t^{16} + 23939953545023043820812t^{15} + 11277023478031223543514t^{14} + 5116714392140870044200t^{13} + 1787307718964572852842t^{12} + 444666453995341989282t^{11} + 77159298583074826193t^{10} + 9242500553122504200t^{9} + 763955384925748848t^{8} + 45651560220254632t^{7} + 2193009207250180t^{6} + 76311012567183t^{5} + 742697066728t^{4} - 35293523906t^{3} - 1001299662t^{2} - 146608t + 18592 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Fl(1, 4, 5)

Show multiplicities

Sector dimension: 14
Spectral radius: $13.5595263$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$13.55953$	1	1
$2.220237 - 6.546742\mathrm{i}$	1	1
$2.220237 + 6.546742\mathrm{i}$	1	1
$-4.206542 - 3.383479\mathrm{i}$	2	1, 1
$-4.206542 + 3.383479\mathrm{i}$	2	1, 1
$-2$	2	2
$-1.293458 - 0.6481502\mathrm{i}$	2	1, 1
$-1.293458 + 0.6481502\mathrm{i}$	2	1, 1
$-1$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right)^{2} \left(\lambda^{3} - 18 \lambda^{2} + 108 \lambda - 648\right) \left(\lambda^{4} + 11 \lambda^{3} + 53 \lambda^{2} + 93 \lambda + 61\right)^{2}\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 6 & 6 & 12 & 12 & 12 & 24 & 18 & 18 & 80 & 56 & 56 & 180 \\ 1 & 0 & 1 & 1 & 4 & 4 & -4 & 12 & 6 & 6 & -6 & -6 & -6 & -24 \\ 1 & 0 & -1 & 2 & 0 & 2 & 2 & 4 & 2 & 4 & 21 & 18 & 12 & 48 \\ 1 & 0 & 2 & -1 & 2 & 0 & 2 & 4 & 4 & 2 & 21 & 12 & 18 & 48 \\ 0 & 1 & 1 & 0 & -1 & 2 & -1 & 1 & 0 & 2 & 2 & 2 & -2 & -9 \\ 0 & 1 & 0 & 1 & 2 & -1 & -1 & 1 & 2 & 0 & 2 & -2 & 2 & -9 \\ 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 12 & 8 & 8 & 18 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 9 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & -1 & 1 & 2 & 0 & 4 & 9 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & -1 & 2 & 4 & 0 & 9 \\ 0 & 0 & 0 & 0 & -\frac{1}{2} & -\frac{1}{2} & -3 & 6 & \frac{5}{2} & \frac{5}{2} & -5 & -3 & -3 & -18 \\ 0 & 0 & 0 & 0 & \frac{11}{4} & \frac{5}{4} & 3 & -2 & \frac{5}{4} & -\frac{5}{4} & 5 & \frac{3}{2} & \frac{9}{2} & 18 \\ 0 & 0 & 0 & 0 & \frac{5}{4} & \frac{11}{4} & 3 & -2 & -\frac{5}{4} & \frac{5}{4} & 5 & \frac{9}{2} & \frac{3}{2} & 18 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{5}{3} & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta