Family 3-141-35-a

This is a Fano fourfold of K3 type, appearing as case C-15 in Bernardara–Fatighenti–Manivel–Tanturri.

Fl(1, 2, 6) · 52.2Z (preferred model)

Defining data

Ambient: Fl(1, 2, 6)
Bundle: $E^{ω_{3}} \oplus E^{ω_{1} + ω_{2}}$ E^(w3) + E^(w1 + w2)
Bundle: $Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (1, 1)$ Q_1^v ot U_2^v ot U_3^v + O(1,1)
ZL62 label: 52.2Z

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	35
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	141
$\chi(X, \mathrm{T}_X)$	=	-18

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	141
$\mathrm{c}_1^2\mathrm{c}_2$	=	126
$\mathrm{c}_2^2$	=	120
$\mathrm{c}_1\mathrm{c}_3$	=	30
$\mathrm{c}_4$	=	33

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		1		23		1		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 + x₂ + 18	0	x₀ + 18	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 20	0	0	x₁	20
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 34	0	0	x₁ - x₂ - 1	0	35
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 30t + 79t^{2} + 30t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(433)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(492)

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( 15t - 1 \right) \left( t + 1 \right) ^{4} \left( 28t^{2} + 2t + 1 \right) \left( 63t^{2} + 15t + 1 \right) \left( 22969128168t^{8} + 85726792800t^{7} + 105519528696t^{6} + 56056690332t^{5} + 15602335260t^{4} + 1990390912t^{3} + 8961457t^{2} + 2522603t - 22184 \right)D^{6} + \left( t + 1 \right) ^{3} \left( 12763025757830880t^{14} + 57882342305992176t^{13} + 97933150613707416t^{12} + 84079209003597864t^{11} + 42402499820380296t^{10} + 13395154453364424t^{9} + 2624218789823232t^{8} + 306992345863830t^{7} + 24162094864453t^{6} + 1300293915389t^{5} + 137214456736t^{4} + 12885212307t^{3} + 10339965t^{2} + 10334436t - 66552 \right)D^{5} + \left( t + 1 \right) ^{2} \left( 106358547981924000t^{15} + 551276566472887920t^{14} + 1114973968782655800t^{13} + 1184225599484427312t^{12} + 748948512244205448t^{11} + 300625495814145780t^{10} + 77020346929383360t^{9} + 11843826329477706t^{8} + 977824595538135t^{7} + 47147582682487t^{6} + 579572924309t^{5} - 70109920020t^{4} - 17191224185t^{3} - 35669775t^{2} - 8822089t + 44368 \right)D^{4} + t \left( t + 1 \right) \left( 446705901524080800t^{15} + 2638532528315606160t^{14} + 6297653982584308680t^{13} + 8115007940892120600t^{12} + 6344652137624336880t^{11} + 3197388815841817680t^{10} + 1061771830887218040t^{9} + 226002748043227482t^{8} + 28038812683363703t^{7} + 1734893290324439t^{6} + 69610039020413t^{5} + 2523443040784t^{4} + 132364195361t^{3} - 1605183879t^{2} - 42019091t + 75520 \right)D^{3} + 2 t \left( 493503662636127360t^{16} + 3304037436536977056t^{15} + 9193407799816125648t^{14} + 14143288135615784208t^{13} + 13462111080646270560t^{12} + 8395959545611492128t^{11} + 3527385397431122628t^{10} + 994885827670206054t^{9} + 179270421094702271t^{8} + 18377397217697256t^{7} + 921938258789156t^{6} + 39147283007231t^{5} + 1883763403684t^{4} + 43915611700t^{3} - 1443982321t^{2} - 7546555t + 11092 \right)D^{2} + 4 t ^{2} \left( 268023540914448480t^{15} + 1752876618499013616t^{14} + 4727617188249030264t^{13} + 7004812231147543920t^{12} + 6389280932534268264t^{11} + 3804015984430645656t^{10} + 1518907082251845300t^{9} + 403200182206678944t^{8} + 66911917126161732t^{7} + 6069147372640296t^{6} + 271295000623096t^{5} + 13801441981508t^{4} + 682776496921t^{3} + 3446382156t^{2} - 460253007t - 487296 \right)D + 96 t ^{3} \left( 4558223484939600t^{14} + 29333341716918120t^{13} + 77386884306529140t^{12} + 111577969350024408t^{11} + 98604112238653860t^{10} + 56674542673249812t^{9} + 21749276461072674t^{8} + 5495043558268842t^{7} + 849356099842551t^{6} + 69002503612812t^{5} + 2941043511749t^{4} + 177650024278t^{3} + 8037562172t^{2} - 49370136t - 4492260 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6)

Show multiplicities

Sector dimension: 9
Spectral radius: $15$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$15$	1	1
$-7.5 - 2.598076\mathrm{i}$	1	1
$-7.5 + 2.598076\mathrm{i}$	1	1
$-1 - 5.196152\mathrm{i}$	1	1
$-1 + 5.196152\mathrm{i}$	1	1
$-1$	4	2, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{4} \left(\lambda - 15\right) \left(\lambda^{2} + 2 \lambda + 28\right) \left(\lambda^{2} + 15 \lambda + 63\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{2} \left(\lambda - 15\right) \left(\lambda^{2} + 2 \lambda + 28\right) \left(\lambda^{2} + 15 \lambda + 63\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 18 & 30 & 48 & 112 & 96 & 195 \\ 1 & -1 & 2 & 0 & 8 & 4 & 39 & 42 & 48 \\ 1 & 2 & -1 & 4 & 10 & 10 & 9 & 12 & 48 \\ 0 & 1 & 2 & -1 & 1 & -3 & -6 & -8 & -9 \\ 0 & 2 & 1 & 1 & -1 & 3 & 4 & 2 & 18 \\ 0 & 0 & 1 & 0 & 2 & 0 & 10 & 10 & 9 \\ 0 & 0 & 0 & 1 & \frac{23}{5} & \frac{9}{5} & -\frac{1}{5} & \frac{12}{5} & \frac{24}{5} \\ 0 & 0 & 0 & 1 & -\frac{6}{5} & \frac{12}{5} & \frac{7}{5} & -\frac{9}{5} & \frac{12}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta