Family 3-136-34-b

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

ℙ²P2 × Fl(1, 3, 4) · 2034.hca (preferred model)

Defining data

Ambient: ℙ²P2 × Fl(1, 3, 4)
Bundle: $E^{ω_{3} + ω_{5}} \oplus E^{ω_{1} + ω_{3}} \oplus E^{ω_{1} + ω_{5}}$ E^(w3 + w5) + E^(w1 + w3) + E^(w1 + w5)
Bundle: $O_{2} (1, 1) \oplus O_{1} (1) ⊠ O_{2} (1, 0) \oplus O_{1} (1) ⊠ O_{2} (0, 1)$ O_2(1,1) + O_1(1) box O_2(1,0) + O_1(1) box O_2(0,1)
ZL62 label: 2034.hca

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	34
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	136
$\chi(X, \mathrm{T}_X)$	=	-12

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	136
$\mathrm{c}_1^2\mathrm{c}_2$	=	124
$\mathrm{c}_2^2$	=	114
$\mathrm{c}_1\mathrm{c}_3$	=	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 + 25	0	0	x₁ + 11	14
$\mathrm{HH}^{ 4 }$ = 34	0	0	0	0	34
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 29t + 76t^{2} + 29t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(489)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(539)

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( 2t + 1 \right) ^{2} \left( 3t + 1 \right) ^{2} \left( 79t^{3} + 51t^{2} + 10t + 1 \right) \left( 244t^{3} - 24t^{2} + 17t - 1 \right) \left( 791779634694872t^{16} + 2998618750813940t^{15} + 3824409847106094t^{14} - 68316446734565t^{13} - 5680198127169466t^{12} - 7398100202207405t^{11} - 5029642952582682t^{10} - 2118269034396188t^{9} - 573980742458837t^{8} - 98562989194339t^{7} - 10102008952473t^{6} - 569957750352t^{5} - 25774911447t^{4} - 2471111530t^{3} - 113509961t^{2} + 1967639t - 10676 \right)D^{6} + \left( 2t + 1 \right) \left( 3t + 1 \right) \left( 1923055374035672436672t^{24} + 9519493455430416969088t^{23} + 18459241682605505274416t^{22} + 12992063426534428358672t^{21} - 12315401968693887465796t^{20} - 38743056110683906579192t^{19} - 45880623581657111335789t^{18} - 34417419362691068646159t^{17} - 18346525741541802462792t^{16} - 7299187154865269505093t^{15} - 2225211351777470350409t^{14} - 527250496327161275619t^{13} - 97509627248247149239t^{12} - 14001203008174871100t^{11} - 1553308307323511009t^{10} - 139576889321125606t^{9} - 12458801005671538t^{8} - 1290724482062135t^{7} - 113944263135537t^{6} - 5754645927565t^{5} - 145193456439t^{4} - 9088304040t^{3} - 494361796t^{2} + 6052381t - 21352 \right)D^{5} + \left( 96152768701783621833600t^{26} + 539368416636681670351680t^{25} + 1229342356307119537593088t^{24} + 1203166606357298321382816t^{23} - 368389394774191309104656t^{22} - 2703669209250115285803520t^{21} - 4059466518756033915678498t^{20} - 3707370146901345382747137t^{19} - 2390956021130684016153916t^{18} - 1150965575508758913730379t^{17} - 424704498210944306453643t^{16} - 121742855716964726172628t^{15} - 27257781765080498827281t^{14} - 4764636996492217356570t^{13} - 646707139492369212042t^{12} - 67577118980513544424t^{11} - 5411172529223567440t^{10} - 337811962643544804t^{9} - 17037513534814041t^{8} - 582184951138556t^{7} + 11510716017505t^{6} + 3304568760498t^{5} + 253013940900t^{4} + 15633914089t^{3} + 522521924t^{2} - 2881337t + 10676 \right)D^{4} + t \left( 403841628547491211701120t^{25} + 2210126590699188767695680t^{24} + 4815271069499307350740960t^{23} + 4110388321064485653287568t^{22} - 2907267900173558054334520t^{21} - 12316633335351157998731380t^{20} - 16951210880259385954785246t^{19} - 14603861842765770383738329t^{18} - 8915213172822268118772656t^{17} - 4046883143136954937635347t^{16} - 1397028563051253114467725t^{15} - 370418556299446274143232t^{14} - 75619571145350124694908t^{13} - 11868125552051117837314t^{12} - 1434216594885735515299t^{11} - 136920502468888798741t^{10} - 11398833971395075059t^{9} - 983200564221903600t^{8} - 88550575449221839t^{7} - 6531441201755108t^{6} - 298633890254000t^{5} - 5192610081222t^{4} + 74291425533t^{3} + 103912826t^{2} - 57731265t - 18369 \right)D^{3} + 2 t ^{2} \left( 446148846776276005307904t^{24} + 2394988972803198167258112t^{23} + 5027344662228134134373712t^{22} + 3763808336452223918819800t^{21} - 4284900104232291979038084t^{20} - 14272917777203868399317250t^{19} - 18488885516485381018136320t^{18} - 15236563957922356240271994t^{17} - 8908332304401331352229382t^{16} - 3859429262976063908350383t^{15} - 1263669585522780691604508t^{14} - 315099208439194250772063t^{13} - 59887215047028454351279t^{12} - 8675272368114574865602t^{11} - 971461096930976430418t^{10} - 89940337554893629146t^{9} - 8049362184975067829t^{8} - 767409856938981276t^{7} - 66631412951370933t^{6} - 4164676842072494t^{5} - 148003420118196t^{4} - 1481274862599t^{3} + 33143754809t^{2} - 1109765573t - 18120232 \right)D^{2} + 4 t ^{2} \left( 242304977128494727020672t^{24} + 1282150350530408745336768t^{23} + 2614478027155966336023888t^{22} + 1740461354686061270805592t^{21} - 2732525962023118767856724t^{20} - 7976040794191129867588002t^{19} - 9905142610228345574509918t^{18} - 7898336570829814011750187t^{17} - 4468516489397027634831982t^{16} - 1867504723369897276517024t^{15} - 586890171162699342237439t^{14} - 139544324569395828481036t^{13} - 25111404895948085452709t^{12} - 3432228687542778400493t^{11} - 367717787137944249934t^{10} - 34484724606969108455t^{9} - 3347394719966704524t^{8} - 330205256138009034t^{7} - 26518479753146291t^{6} - 1415751509095234t^{5} - 39610562668701t^{4} - 171621187095t^{3} + 5684039262t^{2} - 431765496t - 2946576 \right)D + 16 t ^{2} \left( 24724997666172931328640t^{24} + 129551038570475979399360t^{23} + 258814255143275093543472t^{22} + 156904507043673081836232t^{21} - 305851060347536534560388t^{20} - 827635615711585221674850t^{19} - 999627601494526755797704t^{18} - 779217759197132445606118t^{17} - 430730909308359904441552t^{16} - 175424966635760361727647t^{15} - 53511432957883924885226t^{14} - 12287973168570123151923t^{13} - 2124869428283638195129t^{12} - 278893816944838578692t^{11} - 29199690051039443148t^{10} - 2818341000187930340t^{9} - 290406280462436281t^{8} - 28666528102697490t^{7} - 2140402772819245t^{6} - 100592179851616t^{5} - 2284221778360t^{4} - 1041480227t^{3} - 10619107t^{2} - 28148373t - 106760 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Fl(1, 3, 4)

Show multiplicities

Sector dimension: 14
Spectral radius: $16.4428745$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$16.44287$	1	1
$-3.790706 - 4.277176\mathrm{i}$	2	1, 1
$-3.790706 + 4.277176\mathrm{i}$	2	1, 1
$0.2785628 - 3.84209\mathrm{i}$	1	1
$0.2785628 + 3.84209\mathrm{i}$	1	1
$-3$	2	2
$-2.418588$	2	1, 1
$-2$	3	1, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 3\right)^{2} \left(\lambda^{3} + 10 \lambda^{2} + 51 \lambda + 79\right)^{2} \left(\lambda^{3} - 17 \lambda^{2} + 24 \lambda - 244\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda^{3} + 10 \lambda^{2} + 51 \lambda + 79\right) \left(\lambda^{3} - 17 \lambda^{2} + 24 \lambda - 244\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 8 & 8 & 30 & 30 & 12 & 42 & 27 & 27 & 152 & 96 & 96 & 90 \\ 1 & -1 & 1 & 1 & 6 & 6 & 0 & 12 & 8 & 8 & 30 & 18 & 18 & 16 \\ 1 & 1 & -1 & 2 & 2 & 6 & 2 & 6 & 2 & 6 & 33 & 24 & 18 & 20 \\ 1 & 1 & 2 & -1 & 6 & 2 & 2 & 6 & 6 & 2 & 33 & 18 & 24 & 20 \\ 0 & 1 & 1 & 0 & -2 & 2 & 0 & 1 & 0 & 2 & 4 & 2 & 2 & 3 \\ 0 & 1 & 0 & 1 & 2 & -2 & 0 & 1 & 2 & 0 & 4 & 2 & 2 & 3 \\ 0 & 1 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 12 & 8 & 8 & 6 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 2 & -1 & 1 & 6 & 2 & 6 & 3 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 2 & 1 & -1 & 6 & 6 & 2 & 3 \\ 0 & 0 & 0 & 0 & -\frac{1}{2} & -\frac{1}{2} & -3 & 10 & 6 & 6 & 3 & 3 & 3 & 0 \\ 0 & 0 & 0 & 0 & \frac{11}{4} & \frac{5}{4} & 3 & -6 & -\frac{5}{2} & -\frac{9}{2} & -1 & -3 & 0 & 2 \\ 0 & 0 & 0 & 0 & \frac{5}{4} & \frac{11}{4} & 3 & -6 & -\frac{9}{2} & -\frac{5}{2} & -1 & 0 & -3 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 5 & 5 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta