Family 2-130-33-a

This is a Fano fourfold of K3 type, appearing as case GM-22 in Bernardara–Fatighenti–Manivel–Tanturri.

Fl(2, 3, 5) · 45.2V0 (preferred model)

Defining data

Ambient: Fl(2, 3, 5)
Bundle: $E^{ω_{1}} \oplus E^{ω_{3}} \oplus E^{ω_{2} + ω_{3}}$ E^(w1) + E^(w3) + E^(w2 + w3)
Bundle: $U_{3}^{\lor} \oplus O (0, 1) \oplus O (1, 1)$ U_3^v + O(0,1) + O(1,1)
ZL62 label: 45.2V0

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	33
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	130
$\chi(X, \mathrm{T}_X)$	=	-22

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	130
$\mathrm{c}_1^2\mathrm{c}_2$	=	124
$\mathrm{c}_2^2$	=	121
$\mathrm{c}_1\mathrm{c}_3$	=	22
$\mathrm{c}_4$	=	31

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		1		23		1		0
	0		0		0		0
		0		2		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ + 22	0	x₀ + 22	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 26	0	0	x₁ + 6	20
$\mathrm{HH}^{ 4 }$ = 34	0	0	1	0	33
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 28t + 72t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(509)
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( t + 1 \right) ^{3} \left( 1139t^{7} + 1168t^{6} + 3295t^{5} + 1483t^{4} + 1907t^{3} + 206t^{2} - 4t - 1 \right) \left( 198543932756t^{14} + 673785654484t^{13} + 1083471017424t^{12} + 3240984714268t^{11} + 7136804680649t^{10} + 7907690802617t^{9} + 4244114580684t^{8} + 741131366529t^{7} - 352050481027t^{6} - 205779539908t^{5} - 35502111236t^{4} + 2304874402t^{3} + 377489428t^{2} + 7590716t + 33670 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 4748972327590764t^{22} + 23539707731772216t^{21} + 62885927896548056t^{20} + 173348317757247700t^{19} + 423493612882193135t^{18} + 798125886117930970t^{17} + 1132834224310757510t^{16} + 1198900581951047081t^{15} + 918986788419618396t^{14} + 484254760447547755t^{13} + 146966250089927371t^{12} - 1421228844077287t^{11} - 26514266958683020t^{10} - 13931264844934693t^{9} - 3641064880677920t^{8} - 382303554280466t^{7} + 37590748871968t^{6} + 6912630214698t^{5} + 238539107150t^{4} + 17897860910t^{3} + 1474185192t^{2} + 22267098t + 67340 \right)D^{5} + \left( t + 1 \right) \left( 39574769396589700t^{23} + 221102805507621420t^{22} + 622916014952208864t^{21} + 1714550794361552904t^{20} + 4473508732012192689t^{19} + 8937841941871525723t^{18} + 13003228721095305996t^{17} + 13829237107226461398t^{16} + 10599519466027490094t^{15} + 5516494936988392695t^{14} + 1555375531184942125t^{13} - 163401557044014589t^{12} - 396718225937734342t^{11} - 202251722010857389t^{10} - 58041604760313481t^{9} - 9268544878555945t^{8} - 153021162796911t^{7} + 217283785531717t^{6} + 27905526101682t^{5} + 1481406854782t^{4} + 12563967724t^{3} - 1409495174t^{2} - 19150348t - 33670 \right)D^{4} + t \left( 166214031465676740t^{23} + 1046546371140355440t^{22} + 3206854620990521660t^{21} + 8931584963296549988t^{20} + 24397751523334037125t^{19} + 52255181214661970684t^{18} + 80624198786949375763t^{17} + 89664584458109610721t^{16} + 71840237370729197068t^{15} + 39747137445872960000t^{14} + 12644216531618139615t^{13} - 442348037790502918t^{12} - 2924553806308934003t^{11} - 1698682921727454920t^{10} - 548306199049584326t^{9} - 105220752160707277t^{8} - 8666028298974138t^{7} + 1302589230696620t^{6} + 415360670843290t^{5} + 40340185707604t^{4} + 1427756068022t^{3} + 24144906084t^{2} + 132361376t + 29526 \right)D^{3} + 2 t ^{2} \left( 183626930000176208t^{22} + 1115360032813743632t^{21} + 3276541130650031784t^{20} + 9278532088783073748t^{19} + 26063511747432726176t^{18} + 54977853682481966480t^{17} + 80395223953579705150t^{16} + 82186063961351612702t^{15} + 58644584195795747848t^{14} + 27288550624695467355t^{13} + 5552515738390023006t^{12} - 2530913905239157152t^{11} - 2752423728695652422t^{10} - 1222192829567620701t^{9} - 317871504667107046t^{8} - 45703602420656688t^{7} - 1066511913381272t^{6} + 1059289308978700t^{5} + 198592944504224t^{4} + 18722612055416t^{3} + 635074550760t^{2} + 9833869944t + 52770900 \right)D^{2} + 4 t ^{2} \left( 99728418879406044t^{22} + 589507810843154304t^{21} + 1678890101510576492t^{20} + 4847274170784436700t^{19} + 13908840376535203811t^{18} + 28897972738063102672t^{17} + 40414715570525688171t^{16} + 38520245899267884835t^{15} + 24853250835185579456t^{14} + 9767148653713261217t^{13} + 868739939048850895t^{12} - 1644966503604267863t^{11} - 1215216477204065853t^{10} - 453005461657327596t^{9} - 97658977633791316t^{8} - 9991333554674341t^{7} + 529172453354856t^{6} + 373213707074782t^{5} + 53325030304950t^{4} + 4947924524450t^{3} + 165783157212t^{2} + 2184562338t + 10505040 \right)D + 24 t ^{2} \left( 6784246182272520t^{22} + 39355827524003520t^{21} + 109736523142871944t^{20} + 322087044311604100t^{19} + 937616844822059038t^{18} + 1924660202731532572t^{17} + 2604493985498714066t^{16} + 2354495766654223354t^{15} + 1401105921465699044t^{14} + 470831679035872000t^{13} - 11654252638803042t^{12} - 113293213968613019t^{11} - 69086954357994232t^{10} - 22781137880022239t^{9} - 4143627373265350t^{8} - 260881320968900t^{7} + 50130366840084t^{6} + 16863611365828t^{5} + 2145692997938t^{4} + 191429608480t^{3} + 6366628120t^{2} + 69916014t + 269360 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5)

Show multiplicities

Sector dimension: 10
Spectral radius: $16.2655516$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$16.26555$	1	1
$-9.820964 - 3.626103\mathrm{i}$	1	1
$-9.820964 + 3.626103\mathrm{i}$	1	1
$-0.1090817 - 1.054903\mathrm{i}$	1	1
$-0.1090817 + 1.054903\mathrm{i}$	1	1
$-1$	3	1, 2
$-0.2027305 - 0.7259257\mathrm{i}$	1	1
$-0.2027305 + 0.7259257\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{7} + 4 \lambda^{6} - 206 \lambda^{5} - 1907 \lambda^{4} - 1483 \lambda^{3} - 3295 \lambda^{2} - 1168 \lambda - 1139\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda^{7} + 4 \lambda^{6} - 206 \lambda^{5} - 1907 \lambda^{4} - 1483 \lambda^{3} - 3295 \lambda^{2} - 1168 \lambda - 1139\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 15 & 42 & 12 & 30 & 56 & 76 & 160 \\ 1 & -1 & 3 & 8 & 2 & 6 & 0 & 24 & 12 & 8 \\ 1 & 1 & -1 & 8 & 18 & 6 & 12 & 6 & 9 & 48 \\ 0 & 1 & 0 & 0 & 3 & 0 & 2 & 0 & 4 & 12 \\ 0 & 1 & 2 & 0 & -1 & 0 & 0 & 8 & 8 & 6 \\ 0 & 2 & 1 & -2 & 2 & -1 & 2 & 0 & 4 & 12 \\ 0 & 0 & 1 & 2 & 0 & 1 & -1 & 4 & 2 & 0 \\ 0 & 0 & 0 & 1 & \frac{21}{4} & \frac{5}{4} & 4 & -1 & \frac{3}{2} & 12 \\ 0 & 0 & 0 & 2 & \frac{1}{2} & \frac{3}{2} & 0 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta