Family 4-192-45-d

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

ℙ²P2 × Fl(1, 5, 6) · 205G.229wAwA (preferred model)

Defining data

Ambient: ℙ²P2 × Fl(1, 5, 6)
Bundle: $(E^{ω_{3}})^{\oplus 2} \oplus E^{2 ω_{3}} \oplus (E^{ω_{2} + ω_{7}})^{\oplus 2}$ 2 copies of E^(w3) + E^(2w3) + 2 copies of E^(w2 + w7)
Bundle: $O_{2} (1, 0)^{\oplus 2} \oplus O_{2} (2, 0) \oplus (Q_{1} ⊠ O_{2} (0, 1))^{\oplus 2}$ 2 copies of O_2(1,0) + O_2(2,0) + 2 copies of Q_1 box O_2(0,1)
ZL62 label: 205G.229wAwA

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5,5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4,4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	45
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	192
$\chi(X, \mathrm{T}_X)$	=	-6

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	192
$\mathrm{c}_1^2\mathrm{c}_2$	=	144
$\mathrm{c}_2^2$	=	102
$\mathrm{c}_1\mathrm{c}_3$	=	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 + 30	0	0	x₁	30
$\mathrm{HH}^{ 4 }$ = 45	0	0	0	0	45
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(405)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(475)

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( 3t + 1 \right) \left( 91t^{4} + 24t^{3} - 22t^{2} - 9t - 1 \right) \left( 1163t^{4} + 120t^{3} - 46t^{2} + 15t - 1 \right) \left( 134810100590055t^{13} + 210268376810652t^{12} + 120791215754842t^{11} + 126715573834756t^{10} + 103020888950204t^{9} + 41318309663588t^{8} + 8007383234412t^{7} + 745785740009t^{6} + 46257195912t^{5} + 5907172509t^{4} + 458916046t^{3} + 7645278t^{2} + 1175t + 544 \right)D^{6} + \left( 898843514672079321345t^{22} + 1978247298637582133271t^{21} + 1650219797876436018318t^{20} + 1273257842026501159316t^{19} + 1157178090881862369760t^{18} + 630301476585793874070t^{17} + 84288351127251910357t^{16} - 87344627076946332663t^{15} - 54831355271475468239t^{14} - 15202429487521415805t^{13} - 2435026179747775646t^{12} - 252688180168968412t^{11} - 22227825720698191t^{10} - 2842122361384613t^{9} - 539195118921740t^{8} - 83991031943788t^{7} - 7498498302088t^{6} - 421515760318t^{5} - 36677138983t^{4} - 2296499445t^{3} - 30862077t^{2} - 6789t - 1088 \right)D^{5} + \left( 7490362622267327677875t^{22} + 16131202288330916757045t^{21} + 13485265035683515400586t^{20} + 11824171661596270907132t^{19} + 11273971708214186711036t^{18} + 6386861710797810404806t^{17} + 1477136174427374983725t^{16} - 242413073194519957810t^{15} - 246922553624347890480t^{14} - 69292798092925419736t^{13} - 10412348851049254185t^{12} - 993351954940794778t^{11} - 80414742944746089t^{10} - 5083309541900212t^{9} + 396923199311911t^{8} + 130913230516568t^{7} + 11301954439358t^{6} + 515583760273t^{5} + 42132644096t^{4} + 2110880679t^{3} + 24882991t^{2} - 5147t + 544 \right)D^{4} + t \left( 31459523013522776247075t^{21} + 66582216449948367012885t^{20} + 55398962503293931351290t^{19} + 52928766802656556145900t^{18} + 51756947582259025259272t^{17} + 29827931464634706699122t^{16} + 8235312618817139145939t^{15} + 227169736791889265523t^{14} - 520130883880179530457t^{13} - 158900589239324210837t^{12} - 23612783010707693824t^{11} - 2479421149994028328t^{10} - 293339684816779229t^{9} - 34693599967082915t^{8} - 2568861033048456t^{7} - 81788679131554t^{6} + 1860107019082t^{5} + 222835962544t^{4} + 3127506357t^{3} + 35123821t^{2} - 985535t - 663 \right)D^{3} + 2 t ^{2} \left( 34755282567320400425340t^{20} + 72569553953824561187616t^{19} + 59932407894653811400332t^{18} + 60835768845989924448508t^{17} + 60400289726066881830526t^{16} + 35048627817519434601646t^{15} + 10438981726215333280197t^{14} + 1051815857020039108593t^{13} - 251645435396826565381t^{12} - 96623448448178467517t^{11} - 14733304273344843140t^{10} - 1790778298254024944t^{9} - 262476078989876361t^{8} - 31103701491899485t^{7} - 1966841134803644t^{6} - 34792265648488t^{5} + 2874119985393t^{4} + 198772718651t^{3} + 1024144604t^{2} - 5918662t - 405376 \right)D^{2} + 4 t ^{2} \left( 18875713808113665748245t^{20} + 39019576219140567529671t^{19} + 31982444866866361180098t^{18} + 33880333542278989385336t^{17} + 33964228833720674155777t^{16} + 19744714614165937814737t^{15} + 6087768693163168732091t^{14} + 816127435923087981989t^{13} - 39129087633056826212t^{12} - 29298099171222196482t^{11} - 4788571708349970359t^{10} - 697803766974708055t^{9} - 118530978132535477t^{8} - 13750806568924057t^{7} - 773928908264011t^{6} - 7281305028125t^{5} + 1293357801613t^{4} + 74378030967t^{3} + 428237276t^{2} - 3329956t - 91392 \right)D + 8 t ^{2} \left( 3852186491451768520050t^{20} + 7908965779368136074390t^{19} + 6443677761166860265188t^{18} + 7022272637109905975136t^{17} + 7082067668958621863475t^{16} + 4116702318723190331839t^{15} + 1289829912233640249334t^{14} + 193981192561843646210t^{13} + 2937822739258323875t^{12} - 3417055358677472931t^{11} - 628454906585759081t^{10} - 111264860833311229t^{9} - 20913802655349411t^{8} - 2375584543605689t^{7} - 124054529783461t^{6} - 692994533343t^{5} + 202564702373t^{4} + 10197168177t^{3} + 75458498t^{2} - 326808t - 8704 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Fl(1, 5, 6)

Show multiplicities

Sector dimension: 13
Spectral radius: $12.6893597$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$12.68936$	1	1
$2.600426 - 4.994878\mathrm{i}$	1	1
$2.600426 + 4.994878\mathrm{i}$	1	1
$-4.234552 - 1.940705\mathrm{i}$	2	1, 1
$-4.234552 + 1.940705\mathrm{i}$	2	1, 1
$-3$	1	1
$-2.890212$	1	1
$-2.3305$	2	1, 1
$1.799605$	2	1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 9 \lambda^{3} + 22 \lambda^{2} - 24 \lambda - 91\right)^{2} \left(\lambda^{4} - 15 \lambda^{3} + 46 \lambda^{2} - 120 \lambda - 1163\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 9 \lambda^{3} + 22 \lambda^{2} - 24 \lambda - 91\right) \left(\lambda^{4} - 15 \lambda^{3} + 46 \lambda^{2} - 120 \lambda - 1163\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 8 & 6 & 6 & 12 & 9 & 18 & 12 & 48 & 16 & 32 & 120 \\ 1 & 0 & 0 & 2 & 8 & 0 & -2 & 8 & 0 & 0 & 12 & -6 & -32 \\ 1 & 0 & 0 & 1 & 2 & 2 & 2 & 6 & 4 & 12 & 6 & 9 & 32 \\ 1 & 0 & 2 & -1 & 0 & 4 & 2 & 4 & 4 & 12 & 0 & 6 & 32 \\ 0 & 1 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 0 & 4 & -2 & -12 \\ 0 & 1 & 0 & 3 & 2 & -1 & 0 & 4 & 0 & 4 & 4 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 2 & 0 & -2 & 0 & 8 & 0 & 8 & 24 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & -1 & 2 & 4 & 0 & 2 & 12 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & -2 & 2 & 2 & 2 & 6 \\ 0 & 0 & 0 & 0 & 1 & \frac{5}{2} & \frac{1}{2} & \frac{7}{2} & \frac{1}{2} & 1 & 2 & 1 & 6 \\ 0 & 0 & 0 & 0 & 1 & -\frac{3}{2} & -\frac{1}{2} & \frac{3}{2} & \frac{5}{2} & -1 & -2 & -1 & -2 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & -2 & -1 & 2 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{3}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta