Family 2-161-39-a

This is a Fano fourfold of K3 type, appearing as case C-5 in Bernardara–Fatighenti–Manivel–Tanturri.

Fl(2, 3, 8) · 705.66w6w1 (preferred model)

Defining data

Ambient: Fl(2, 3, 8)
Bundle: $(E^{ω_{7}})^{\oplus 2} \oplus E^{ω_{1} + ω_{3}} \oplus E^{ω_{2} + ω_{3}}$ 2 copies of E^(w7) + E^(w1 + w3) + E^(w2 + w3)
Bundle: $(⋀^{4} Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor})^{\oplus 2} \oplus U_{2}^{\lor} \otimes Σ_{(2, 1)} U_{3}^{\lor} \oplus O (1, 1)$ 2 copies of wedge^4 Q_1^v ot U_2^v ot wedge^2 U_3^v + U_2^v ot Sigma_(2,1) U_3^v + O(1,1)
ZL62 label: 705.66w6w1

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,6) \\ \det\mathcal{E} &\cong \mathcal{O}(2,5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	39
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	161
$\chi(X, \mathrm{T}_X)$	=	-21

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	161
$\mathrm{c}_1^2\mathrm{c}_2$	=	134
$\mathrm{c}_2^2$	=	118
$\mathrm{c}_1\mathrm{c}_3$	=	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 + 21	0	x₀ + 21	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 31	0	0	x₁ - 2	33
$\mathrm{HH}^{ 4 }$ = 40	0	0	1	0	39
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 34t + 91t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(407)
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) \left( 8281t^{5} - 3756t^{4} + 910t^{3} + 131t^{2} - t - 1 \right) \left( 190212426768t^{10} - 691335117260t^{9} - 675810865359t^{8} - 1789397104282t^{7} + 695942554268t^{6} - 228291050046t^{5} - 14830543430t^{4} + 1368283162t^{3} - 114036780t^{2} + 131402t - 12075 \right)D^{6} + 2 \left( 9450894636394848t^{16} - 33769304764694590t^{15} - 54334728890635359t^{14} - 121506352123636790t^{13} + 6931959911269821t^{12} + 20082773264407456t^{11} - 18452512355659242t^{10} + 4119014168434053t^{9} - 358422140525445t^{8} - 91957664438262t^{7} + 7107512302661t^{6} - 1039636257324t^{5} - 71916799109t^{4} + 3400563180t^{3} - 232914212t^{2} + 221253t - 12075 \right)D^{5} + \left( 91358648151816864t^{16} - 360730115138967392t^{15} - 520501438235625547t^{14} - 1386884082522094285t^{13} + 215812892614209616t^{12} + 52708108367230999t^{11} - 126057316888441920t^{10} + 17964623272879502t^{9} + 234089273001103t^{8} - 266935307088370t^{7} + 30536603596901t^{6} - 327161663753t^{5} + 63682013670t^{4} - 7017663669t^{3} + 445804694t^{2} + 1309496t + 12075 \right)D^{4} + 2 t \left( 113410735636738176t^{15} - 480608791325501566t^{14} - 645114460943533012t^{13} - 1957284847749903189t^{12} + 472207982153546219t^{11} - 97598158484765445t^{10} - 125866293663983616t^{9} + 6493917062523671t^{8} + 2809571830457666t^{7} - 258129445464972t^{6} + 34443455040636t^{5} - 3216009263727t^{4} - 28012384533t^{3} - 808818473t^{2} - 15380704t - 1275 \right)D^{3} + 4 t ^{2} \left( 76000944367675236t^{14} - 338895380992478219t^{13} - 433687640059929270t^{12} - 1452271205349139181t^{11} + 445333093901867959t^{10} - 156665863021897518t^{9} - 74079831498609493t^{8} - 205723759638866t^{7} + 2534516405809130t^{6} - 167968331081039t^{5} + 11849145889968t^{4} - 1662965192537t^{3} - 5333880573t^{2} - 571179904t - 5808397 \right)D^{2} + 32 t ^{2} \left( 6497490062521458t^{14} - 30060097567036537t^{13} - 37283405416990979t^{12} - 134417630753527828t^{11} + 47383034576605921t^{10} - 19522101700585285t^{9} - 5896759418825271t^{8} - 122934695504162t^{7} + 222495485638444t^{6} - 15619967431188t^{5} + 345985383676t^{4} - 106587075636t^{3} - 273377641t^{2} - 45889492t - 289800 \right)D + 48 t ^{2} \left( 1181361829549356t^{14} - 5614655007120341t^{13} - 6821861016426757t^{12} - 25984354444300805t^{11} + 9970050079665512t^{10} - 4401680816615087t^{9} - 1035873025323330t^{8} - 16709542409570t^{7} + 37519239644092t^{6} - 3080378074503t^{5} - 18582885381t^{4} - 14571753365t^{3} - 68086796t^{2} - 7638825t - 32200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 8)

Show multiplicities

Sector dimension: 9
Spectral radius: $13.0182367$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$13.01824$	1	1
$-8.682916 - 5.299648\mathrm{i}$	1	1
$-8.682916 + 5.299648\mathrm{i}$	1	1
$1.673797 - 1.829098\mathrm{i}$	1	1
$1.673797 + 1.829098\mathrm{i}$	1	1
$-1$	1	1
$0$	3	1, 2

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 1\right) \left(\lambda^{5} + \lambda^{4} - 131 \lambda^{3} - 910 \lambda^{2} + 3756 \lambda - 8281\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right) \left(\lambda^{5} + \lambda^{4} - 131 \lambda^{3} - 910 \lambda^{2} + 3756 \lambda - 8281\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 24 & 36 & 18 & 0 & 32 & 160 \\ 1 & -1 & 0 & 8 & 20 & 12 & 0 & 0 & 32 \\ 1 & 2 & 0 & -8 & -4 & 4 & 60 & 90 & -32 \\ 0 & 1 & 0 & 0 & 0 & 1 & 16 & 24 & 0 \\ 0 & 2 & 1 & -2 & -1 & -1 & 0 & 8 & 0 \\ 0 & 0 & 1 & 4 & 2 & 1 & 0 & 0 & 24 \\ 0 & 0 & 0 & 2 & 1 & -\frac{3}{8} & -\frac{3}{2} & -\frac{3}{4} & 8 \\ 0 & 0 & 0 & 0 & 2 & \frac{7}{4} & 1 & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 4 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta