Family 3-358-76-a

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

Defining data

Ambient: Fl(1, 2, 4) × ℙ²P2
Bundle: $E^{ω_{1}} \oplus E^{ω_{3} + ω_{4}}$ E^(w1) + E^(w3 + w4)
Bundle: $O_{1} (1, 0) \oplus Q_{1, 1}^{\lor} \otimes U_{1, 2}^{\lor} \otimes U_{1, 3}^{\lor} ⊠ O_{2} (1)$ O_1(1,0) + Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1)
ZL62 label: 2032.2c

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	76
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	358
$\chi(X, \mathrm{T}_X)$	=	9

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	358
$\mathrm{c}_1^2\mathrm{c}_2$	=	184
$\mathrm{c}_2^2$	=	99
$\mathrm{c}_1\mathrm{c}_3$	=	58
$\mathrm{c}_4$	=	13

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		5		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 - 9	0	x₀ - 9	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 55	0	0	x₁ - 46	101
$\mathrm{HH}^{ 4 }$ = 76	0	0	0	0	76
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 71t + 214t^{2} + 71t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(20)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	No match

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( 4t^{2} + 1 \right) \left( 7t^{2} - t + 1 \right) \left( 2912t^{5} + 1536t^{4} + 240t^{3} - 4t^{2} - 2t - 1 \right) \left( 528665580713984t^{17} + 3698839526889472t^{16} + 6934544791387136t^{15} + 6646836159916544t^{14} + 3327637196324352t^{13} + 381162644655744t^{12} - 528554267774304t^{11} - 303894959726784t^{10} - 66248051368408t^{9} - 5587803428096t^{8} - 285054291550t^{7} + 107191924798t^{6} + 64870760796t^{5} + 11567473640t^{4} + 897032293t^{3} + 49682237t^{2} + 314938t + 11098 \right)D^{6} + \left( 3620843250284013551616t^{27} + 28399953493448057421824t^{26} + 66910402437937776033792t^{25} + 88077306343857202462720t^{24} + 74668906296614845644800t^{23} + 41973470930628187013120t^{22} + 14415462133587466600448t^{21} + 1164635839673358327808t^{20} - 2230187136596518160384t^{19} - 1796212631770494423040t^{18} - 811082381912115087872t^{17} - 251969735996818994432t^{16} - 57286885587509557888t^{15} - 9261831446512759232t^{14} - 779147643602468160t^{13} + 69027477036149376t^{12} + 40312548519876328t^{11} + 11002026739331340t^{10} + 2455576672106650t^{9} + 389672159585810t^{8} + 38921482608790t^{7} + 3858050272702t^{6} + 603702985973t^{5} + 71336678689t^{4} + 4477702255t^{3} + 196022262t^{2} + 833834t + 22196 \right)D^{5} + \left( 30173693752366779596800t^{27} + 241790745576883210158080t^{26} + 568962281902608079847424t^{25} + 731162747637362885132288t^{24} + 581724505619946704289792t^{23} + 280517972238295955447808t^{22} + 57672254690858916798464t^{21} - 23231711638312850874368t^{20} - 27594046657285325859840t^{19} - 14579252968975508698112t^{18} - 5164818510648217180160t^{17} - 1257700204528676068480t^{16} - 204494121738026057728t^{15} - 16546515097599977632t^{14} + 2767435083989109072t^{13} + 1690921304216263656t^{12} + 509792052963479016t^{11} + 114262567599538376t^{10} + 18651538696358846t^{9} + 2123997950916236t^{8} + 174456081461876t^{7} + 8529850521714t^{6} - 350962175353t^{5} - 83923746840t^{4} - 4260912170t^{3} - 197088259t^{2} - 820588t - 11098 \right)D^{4} + t \left( 126729513759940474306560t^{26} + 1032431870756802497085440t^{25} + 2429914427358206775132160t^{24} + 3074093641774262771908608t^{23} + 2334267822480647279837184t^{22} + 980503846111176198340608t^{21} + 66526661569637669412864t^{20} - 180227772462425918631936t^{19} - 129060201899855860955136t^{18} - 53488519514991043323904t^{17} - 15524114535763409317376t^{16} - 2972212242662730299648t^{15} - 289006542737636142336t^{14} + 30426732577099983616t^{13} + 23716398146249567072t^{12} + 7500587637996867440t^{11} + 1740154869020644904t^{10} + 315898865919834636t^{9} + 44396671219819226t^{8} + 4677121894858022t^{7} + 365526848125038t^{6} + 20953685503854t^{5} + 658115851033t^{4} + 9831938267t^{3} + 216124373t^{2} + 612530t + 2046 \right)D^{3} + 2 t ^{2} \left( 140005939010981857329152t^{25} + 1154885001177829473255424t^{24} + 2720211728276044238716928t^{23} + 3407187922267968198836224t^{22} + 2502833057707281203666944t^{21} + 934913803714999085625344t^{20} - 58554394242237042499584t^{19} - 255993213976045218115584t^{18} - 146178259376647120476672t^{17} - 50285008419962480327168t^{16} - 12091987393574544337408t^{15} - 1754842832415228029120t^{14} - 25136705776583917504t^{13} + 70144544108286478800t^{12} + 28511371273486421160t^{11} + 7949152337140176916t^{10} + 1685670957003425664t^{9} + 273613782009710670t^{8} + 33936634593465098t^{7} + 3194018071964014t^{6} + 229280768864846t^{5} + 12328830804415t^{4} + 363653327459t^{3} + 6912454267t^{2} + 142016293t + 185802 \right)D^{2} + 8 t ^{2} \left( 38018854127982142291968t^{25} + 316455423462111834406912t^{24} + 746023166141940350304256t^{23} + 928627600627687668391936t^{22} + 666822046330242676170752t^{21} + 226919333944186748221440t^{20} - 40093687226363839465472t^{19} - 79261014148146221483520t^{18} - 39546456129320432007936t^{17} - 11696020049162786632448t^{16} - 2333680318910832944576t^{15} - 228807480523504558688t^{14} + 33050360007924824912t^{13} + 22921306673217882496t^{12} + 7861151883773232968t^{11} + 2079181637820989800t^{10} + 414190909005464362t^{9} + 61794910011914544t^{8} + 6949761727876306t^{7} + 594593292914061t^{6} + 39346185110115t^{5} + 1970298629821t^{4} + 54459302513t^{3} + 1222551953t^{2} + 21758652t + 11098 \right)D + 48 t ^{3} \left( 2586316607345723965440t^{24} + 21658306012681486991360t^{23} + 51093992821437005791232t^{22} + 63362647187482019471360t^{21} + 44837762845759624294400t^{20} + 14268799563771111583744t^{19} - 3783580103710517293056t^{18} - 5798593814428507908608t^{17} - 2661429860463037215232t^{16} - 701091258880005272320t^{15} - 117020632334267727680t^{14} - 5679946059578112768t^{13} + 3654870297535905008t^{12} + 1675802241651599368t^{11} + 536905564204209748t^{10} + 138138254681748774t^{9} + 26335019843458240t^{8} + 3700271869869836t^{7} + 387455273389742t^{6} + 30737719685849t^{5} + 1891211742412t^{4} + 88550182968t^{3} + 2326973534t^{2} + 59027976t + 898938 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 4) × ℙ²P2

Show multiplicities

Sector dimension: 13
Spectral radius: $7.21615874$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$7.216159$	1	1
$-1.726014 - 6.140911\mathrm{i}$	1	1
$-1.726014 + 6.140911\mathrm{i}$	1	1
$-4$	2	1, 1
$-2.882066 - 1.269297\mathrm{i}$	1	1
$-2.882066 + 1.269297\mathrm{i}$	1	1
$0.5 - 2.598076\mathrm{i}$	2	1, 1
$0.5 + 2.598076\mathrm{i}$	2	1, 1
$0 - 2\mathrm{i}$	1	1
$0 + 2\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda^{2} + 4\right) \left(\lambda^{2} - \lambda + 7\right)^{2} \left(\lambda^{5} + 2 \lambda^{4} + 4 \lambda^{3} - 240 \lambda^{2} - 1536 \lambda - 2912\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda^{2} + 4\right) \left(\lambda^{2} - \lambda + 7\right) \left(\lambda^{5} + 2 \lambda^{4} + 4 \lambda^{3} - 240 \lambda^{2} - 1536 \lambda - 2912\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 6 & 4 & 4 & 4 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 & 3 & 0 \\ 2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 4 \\ 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 & 3 & 0 & 3 & 0 \\ 0 & 2 & 3 & -1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & -2 & -3 \\ 0 & 1 & 0 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 3 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 1 & 3 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 2 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 3 & 3 & 1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 3 & 0 & 4 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 4 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta