Family 4-172-41-d

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

ℙ¹P1 × ℙ¹P1 × Fl(1, 3, 4) · 1134.hfa (preferred model)

Defining data

Ambient: ℙ¹P1 × ℙ¹P1 × Fl(1, 3, 4)
Bundle: $E^{ω_{3} + ω_{5}} \oplus E^{ω_{2} + ω_{3}} \oplus E^{ω_{1} + ω_{5}}$ E^(w3 + w5) + E^(w2 + w3) + E^(w1 + w5)
Bundle: $O_{3} (1, 1) \oplus O_{2} (1) ⊠ O_{3} (1, 0) \oplus O_{1} (1) ⊠ O_{3} (0, 1)$ O_3(1,1) + O_2(1) box O_3(1,0) + O_1(1) box O_3(0,1)
ZL62 label: 1134.hfa

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;2,2) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	41
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	172
$\chi(X, \mathrm{T}_X)$	=	-6

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	172
$\mathrm{c}_1^2\mathrm{c}_2$	=	136
$\mathrm{c}_2^2$	=	106
$\mathrm{c}_1\mathrm{c}_3$	=	52
$\mathrm{c}_4$	=	22

Hodge diamond

				1
			0		0
		0		4		0
	0		0		0		0
0		0		12		0		0
	0		0		0		0
		0		4		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ + 6	0	x₀ + 6	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 26	0	0	x₁ + 4	22
$\mathrm{HH}^{ 4 }$ = 41	0	0	0	0	41
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 36t + 98t^{2} + 36t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(471)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(526)

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) \left( 6t + 1 \right) \left( 14t - 1 \right) \left( 2t + 1 \right) ^{3} \left( 416t^{4} + 144t^{3} + 8t^{2} - 4t - 1 \right) \left( 600112383721472t^{15} + 1996467209043968t^{14} + 2893287509067776t^{13} + 2363112536049664t^{12} + 1157177022450304t^{11} + 312921761466816t^{10} + 16403415218320t^{9} - 20792143134800t^{8} - 8545842230660t^{7} - 1719890872572t^{6} - 201561710788t^{5} - 13381302832t^{4} - 419704442t^{3} - 1394333t^{2} + 208448t - 2881 \right)D^{6} + 2 \left( 2t + 1 \right) ^{2} \left( 1761507479488101875712t^{23} + 7586540243426373271552t^{22} + 14815972980336924557312t^{21} + 17246424072095647137792t^{20} + 13171561593574091980800t^{19} + 6771219775166328619008t^{18} + 2233023716728479211520t^{17} + 342057307706059155456t^{16} - 74630876344917514240t^{15} - 62871056972355881472t^{14} - 19739516117770681344t^{13} - 3672913763162457792t^{12} - 384739109495033696t^{11} - 5489970879503040t^{10} + 5330662918099504t^{9} + 1012608613618812t^{8} + 115404560234002t^{7} + 10308169167500t^{6} + 743411024138t^{5} + 37200300326t^{4} + 987529107t^{3} + 5457822t^{2} - 335720t + 2881 \right)D^{5} + \left( 2t + 1 \right) \left( 58716915982936729190400t^{24} + 272314391834339343073280t^{23} + 575968522812509952933888t^{22} + 731685413950528862814208t^{21} + 616712909233443476930560t^{20} + 356701577554316129976320t^{19} + 138306685167311551623168t^{18} + 30300141191329342784512t^{17} - 901850303530335962624t^{16} - 3537205647446993741184t^{15} - 1492378314416500634176t^{14} - 372350671581984333536t^{13} - 62100505943296532112t^{12} - 6822281814123559296t^{11} - 418765840995955744t^{10} - 538180301153320t^{9} + 2059059592314480t^{8} + 97941259576716t^{7} - 10826989197108t^{6} - 1772159987456t^{5} - 107412225704t^{4} - 2968704046t^{3} - 28557783t^{2} + 343866t - 2881 \right)D^{4} + 2 t \left( 246611047128334262599680t^{24} + 1234261124172965541314560t^{23} + 2831973963033328180264960t^{22} + 3929224010221987633823744t^{21} + 3651659555719239474348032t^{20} + 2364808256010764447678464t^{19} + 1059194293099529840541696t^{18} + 297180695853349647820800t^{17} + 23845325607470899747840t^{16} - 22432221493672154956032t^{15} - 13203378213562495666944t^{14} - 4135253436168112424320t^{13} - 878848017429860052736t^{12} - 133040812047816341744t^{11} - 14267754751084995808t^{10} - 1031016006038871200t^{9} - 42875043509973888t^{8} - 265982081475012t^{7} + 73820446549890t^{6} + 3909644859052t^{5} + 80022566710t^{4} + 1672026026t^{3} + 166081471t^{2} + 5072908t + 1943 \right)D^{3} + 4 t ^{2} \left( 272446490160826423443456t^{23} + 1335870575255665443340288t^{22} + 2996627295798209551532032t^{21} + 4052735122297563304951808t^{20} + 3654707911413507972300800t^{19} + 2278123325957111112916992t^{18} + 964553377771366397782016t^{17} + 240082955304751047799296t^{16} + 1573024203082647531520t^{15} - 27953750074560589496000t^{14} - 13857233256704513540096t^{13} - 4052249589476153213440t^{12} - 824198486203368321856t^{11} - 121111793772022194988t^{10} - 12835599837814623360t^{9} - 951876752500316100t^{8} - 45695760151087446t^{7} - 1093833280255580t^{6} + 12382923031504t^{5} + 1858339507432t^{4} + 74690304309t^{3} + 3083028616t^{2} + 116696990t + 1773535 \right)D^{2} + 8 t ^{2} \left( 147966628277000557559808t^{23} + 714493729541507982032896t^{22} + 1575572429726270801051648t^{21} + 2089285274973444923981824t^{20} + 1839879611917954431156224t^{19} + 1111749164607572134551552t^{18} + 448424841028685935785984t^{17} + 99064186924973508371968t^{16} - 7406335877480577928192t^{15} - 15971344488017560092736t^{14} - 7143459707159427253504t^{13} - 1985615834344253557856t^{12} - 389092560136527105696t^{11} - 55440316548742098604t^{10} - 5732407416386718512t^{9} - 418996158046757024t^{8} - 20273780211517462t^{7} - 528820655415966t^{6} + 1734572267376t^{5} + 680123927736t^{4} + 30806842202t^{3} + 1295968373t^{2} + 36713435t + 339958 \right)D + 32 t ^{2} \left( 15098635538469444648960t^{23} + 72147670799796110622720t^{22} + 157221313927735208312832t^{21} + 205608819229676032622592t^{20} + 177999080584194128379904t^{19} + 105109174610586062880768t^{18} + 40825779152030866692096t^{17} + 8107914151017720827392t^{16} - 1258106083844131592704t^{15} - 1664295874932109311552t^{14} - 698741632879378500864t^{13} - 187436021092273549680t^{12} - 35700241387043202512t^{11} - 4957350448236443496t^{10} - 500290089942422952t^{9} - 35741214639600914t^{8} - 1689839568520352t^{7} - 42185926753542t^{6} + 288575764903t^{5} + 63130658789t^{4} + 2717478830t^{3} + 103120590t^{2} + 2360611t + 14405 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ¹P1 × ℙ¹P1 × Fl(1, 3, 4)

Show multiplicities

Sector dimension: 18
Spectral radius: $14$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$14$	1	1
$-6$	2	1, 1
$5.327276$	2	1, 1
$-2.781989 - 3.607022\mathrm{i}$	2	1, 1
$-2.781989 + 3.607022\mathrm{i}$	2	1, 1
$-4$	1	1
$-3.763298$	2	1, 1
$-2$	6	1, 1, 1, 1, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{6} \left(\lambda + 4\right) \left(\lambda + 6\right)^{2} \left(\lambda - 14\right) \left(\lambda^{4} + 4 \lambda^{3} - 8 \lambda^{2} - 144 \lambda - 416\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 4\right) \left(\lambda + 6\right) \left(\lambda - 14\right) \left(\lambda^{4} + 4 \lambda^{3} - 8 \lambda^{2} - 144 \lambda - 416\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 8 & 8 & 0 & 9 & 9 & 9 & 9 & 30 & 18 & 18 & 12 & 12 & 36 & 36 & 40 \\ 1 & -1 & 0 & 0 & 1 & 2 & 8 & 6 & 0 & 0 & 6 & 2 & 6 & 9 & 9 & 21 & 9 & 24 \\ 1 & 0 & -1 & 1 & 0 & 2 & 0 & 0 & 6 & 8 & 6 & 6 & 2 & 9 & 9 & 9 & 21 & 24 \\ 1 & 0 & 1 & -1 & 2 & 0 & 2 & 4 & 0 & 0 & 6 & 2 & 6 & 0 & 0 & 6 & 6 & 8 \\ 1 & 1 & 0 & 2 & -1 & 0 & 0 & 0 & 4 & 2 & 6 & 6 & 2 & 0 & 0 & 6 & 6 & 8 \\ 0 & 1 & 1 & 0 & 0 & -2 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 6 & 6 & 6 & 6 & 12 \\ 0 & 1 & 0 & 1 & 0 & 1 & -2 & 2 & 0 & 0 & 1 & 0 & 2 & 0 & 0 & 4 & 0 & 6 \\ 0 & 1 & 0 & 0 & 1 & 0 & 2 & -2 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & 6 & 0 & 6 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -2 & 2 & 0 & 0 & 0 & 2 & 4 & 0 & 6 & 6 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 2 & -2 & 1 & 2 & 0 & 0 & 0 & 0 & 4 & 6 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & -1 & 1 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 2 & 1 & -1 & 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & -\frac{1}{3} & \frac{7}{6} & -\frac{1}{2} & -\frac{1}{2} & \frac{4}{3} & -\frac{5}{3} & -2 & \frac{3}{2} & \frac{5}{6} & \frac{1}{3} & \frac{10}{3} \\ 0 & 0 & 0 & 0 & 0 & 1 & -\frac{1}{2} & \frac{7}{6} & -\frac{1}{3} & 2 & -\frac{1}{2} & -\frac{5}{3} & \frac{4}{3} & \frac{3}{2} & -2 & \frac{1}{3} & \frac{5}{6} & \frac{10}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{3}{2} & 0 & 0 & \frac{5}{2} & 1 & 2 & \frac{1}{2} & 1 & 0 & \frac{1}{2} & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{3}{2} & \frac{5}{2} & 2 & 1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{3}{2} & \frac{5}{2} & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta