Family 2-80-23-d

This is a Fano fourfold of K3 type, appearing as case GM-19 in Bernardara–Fatighenti–Manivel–Tanturri.

ℙ²P2 × Gr(2, 5) · 2041.3mi (preferred model)

Defining data

Ambient: ℙ²P2 × Gr(2, 5)
Bundle: $E^{ω_{4}} \oplus E^{ω_{1} + ω_{4}} \oplus E^{ω_{2} + ω_{4}}$ E^(w4) + E^(w1 + w4) + E^(w2 + w4)
Bundle: $O_{2} (1) \oplus O_{1} (1) ⊠ O_{2} (1) \oplus Q_{1} ⊠ O_{2} (1)$ O_2(1) + O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label: 2041.3mi

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	23
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	80
$\chi(X, \mathrm{T}_X)$	=	-26

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	80
$\mathrm{c}_1^2\mathrm{c}_2$	=	104
$\mathrm{c}_2^2$	=	135
$\mathrm{c}_1\mathrm{c}_3$	=	14
$\mathrm{c}_4$	=	35

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		1		27		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 + 26	0	x₀ + 26	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 16	0	0	x₁ + 12	4
$\mathrm{HH}^{ 4 }$ = 24	0	0	1	0	23
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 18t + 42t^{2} + 18t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(629)
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( 3t + 1 \right) ^{3} \left( 106324t^{6} + 86442t^{5} + 25871t^{4} + 3350t^{3} + 222t^{2} + 13t - 1 \right) \left( 56975047386432t^{14} + 144669421955264t^{13} + 187930396981120t^{12} + 165142258129280t^{11} + 103892507951892t^{10} + 46546986758364t^{9} + 14668893972458t^{8} + 3218422601648t^{7} + 486669534190t^{6} + 50279039416t^{5} + 3546534706t^{4} + 170240604t^{3} + 4841694t^{2} + 25965t + 72 \right)D^{6} + \left( 3t + 1 \right) ^{2} \left( 1526569364455378983936t^{22} + 5665234783985960646912t^{21} + 10474646771041079717504t^{20} + 12865004643141581675072t^{19} + 11614205946882692642880t^{18} + 8010381395953718777616t^{17} + 4283470276470586582248t^{16} + 1787618134451080143372t^{15} + 584485809107104906196t^{14} + 150096000275942321210t^{13} + 30311324999331347984t^{12} + 4814923625614949550t^{11} + 601596106661938062t^{10} + 59200215537651616t^{9} + 4615594864344970t^{8} + 289654841397748t^{7} + 15058643534074t^{6} + 668436835561t^{5} + 25533323988t^{4} + 825099330t^{3} + 19272843t^{2} + 77679t + 144 \right)D^{5} + \left( 3t + 1 \right) \left( 38164234111384474598400t^{23} + 147500382653790432318720t^{22} + 284487207143684959207296t^{21} + 365147311976289319779008t^{20} + 345273526906716763153536t^{19} + 250147119065503874014736t^{18} + 141010694174404337780440t^{17} + 62298914309495604948052t^{16} + 21669816203199763129296t^{15} + 5953154393035592221546t^{14} + 1294154454409080145156t^{13} + 222784381565658613760t^{12} + 30359506839972259018t^{11} + 3269572725789044826t^{10} + 276839255140825484t^{9} + 18133827357781896t^{8} + 876996162166281t^{7} + 27415917319164t^{6} + 296807327645t^{5} - 14613009200t^{4} - 735694017t^{3} - 16140906t^{2} - 56655t - 72 \right)D^{4} + t \left( 480869349803444379939840t^{23} + 1950960811374005494199040t^{22} + 3950084295564314509941120t^{21} + 5323334688359426102010048t^{20} + 5293026207193275366704576t^{19} + 4045237158888764613343600t^{18} + 2416481489411531202988056t^{17} + 1137566629764484619306244t^{16} + 424231825147305063201020t^{15} + 125810668467445387206546t^{14} + 29747622322446541908728t^{13} + 5616328690871726032568t^{12} + 847142848389715205686t^{11} + 102058224871678462910t^{10} + 9803457638314436612t^{9} + 746364890196743872t^{8} + 44271629483983778t^{7} + 1965446896782111t^{6} + 60322194232886t^{5} + 1113195328253t^{4} + 10111596774t^{3} + 55338093t^{2} + 346788t + 261 \right)D^{3} + t ^{2} \left( 1062492277660943772819456t^{22} + 4196022951642055449338880t^{21} + 8313488558373076835083776t^{20} + 11013060684084374459208192t^{19} + 10775833919758887207751808t^{18} + 8084027445622721241870272t^{17} + 4718531968511495682209440t^{16} + 2159071036887450275843616t^{15} + 778682457407788187189368t^{14} + 222307686401126161289260t^{13} + 50402678808095489399408t^{12} + 9095567290526376596688t^{11} + 1308292602999809311660t^{10} + 150074241253538445076t^{9} + 13703557231554449328t^{8} + 987336604827735706t^{7} + 54742787232639828t^{6} + 2211034802345396t^{5} + 58847029046924t^{4} + 889439100497t^{3} + 7976675160t^{2} + 97228755t + 377136 \right)D^{2} + 4 t ^{2} \left( 288521609882066627963904t^{22} + 1116617787583382702847744t^{21} + 2177161528110291258609024t^{20} + 2848173941517441413841216t^{19} + 2753594957174370740294912t^{18} + 2036332537092445905977712t^{17} + 1167089840473829458254968t^{16} + 522127900607299593853660t^{15} + 183359997063898447889752t^{14} + 50785030547943108423592t^{13} + 11135330990573771537436t^{12} + 1938469453862937188672t^{11} + 268501269334193675664t^{10} + 29622961783991990092t^{9} + 2596973209481391504t^{8} + 178729560800471884t^{7} + 9340686428613184t^{6} + 345581210668291t^{5} + 7965053152852t^{4} + 94270340769t^{3} + 750825612t^{2} + 11895462t + 42624 \right)D + 8 t ^{2} \left( 58881961200421760808960t^{22} + 224734901300339292495360t^{21} + 433409848001476919126784t^{20} + 562165041988612179110016t^{19} + 538997388587229651058944t^{18} + 394556266357897941074528t^{17} + 223174431821687803790576t^{16} + 98216416663720201805368t^{15} + 33823407267973195576512t^{14} + 9160269956347077428416t^{13} + 1959045305772385871360t^{12} + 331956322620065806852t^{11} + 44688717055099059704t^{10} + 4786349805748813734t^{9} + 406596455076495300t^{8} + 26980920718532122t^{7} + 1343279181325492t^{6} + 46166224298169t^{5} + 938118891408t^{4} + 8667319359t^{3} + 56755116t^{2} + 924462t + 3168 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Gr(2, 5)

Show multiplicities

Sector dimension: 10
Spectral radius: $27.1669237$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$27.16692$	1	1
$-1.948207 - 9.088851\mathrm{i}$	1	1
$-1.948207 + 9.088851\mathrm{i}$	1	1
$-4$	1	1
$-3.916795$	1	1
$-3.176857 - 1.213354\mathrm{i}$	1	1
$-3.176857 + 1.213354\mathrm{i}$	1	1
$-3$	3	1, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda + 4\right) \left(\lambda^{6} - 13 \lambda^{5} - 222 \lambda^{4} - 3350 \lambda^{3} - 25871 \lambda^{2} - 86442 \lambda - 106324\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda + 4\right) \left(\lambda^{6} - 13 \lambda^{5} - 222 \lambda^{4} - 3350 \lambda^{3} - 25871 \lambda^{2} - 86442 \lambda - 106324\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 32 & 234 & 87 & 156 & 243 & 1136 & 1060 & 4840 \\ 1 & -1 & 6 & 20 & 0 & 24 & 36 & 102 & 90 & 576 \\ 1 & 1 & 2 & 32 & 16 & 20 & 32 & 192 & 195 & 848 \\ 0 & 1 & 1 & 1 & -1 & 4 & 6 & 8 & 0 & 48 \\ 0 & 1 & 0 & 6 & 0 & 2 & 4 & 24 & 20 & 84 \\ 0 & 0 & 1 & 1 & 1 & -3 & 0 & 8 & 10 & 48 \\ 0 & 0 & 1 & 1 & 1 & 2 & 0 & 16 & 20 & 84 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & 3 & \frac{9}{2} & 1 & \frac{5}{2} & 24 \\ 0 & 0 & 0 & 1 & 1 & -\frac{4}{5} & -\frac{6}{5} & \frac{16}{5} & 0 & \frac{32}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{5}{4} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta