Family 2-256-57-b

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

Fl(2, 4, 8) · 709.666A7 (preferred model)

Defining data

Ambient: Fl(2, 4, 8)
Bundle: $(E^{ω_{7}})^{\oplus 3} \oplus E^{2 ω_{1}} \oplus E^{2 ω_{4}}$ 3 copies of E^(w7) + E^(2w1) + E^(2w4)
Bundle: $(⋀^{3} Q_{1}^{\lor} \otimes ⋀^{2} U_{2}^{\lor} \otimes ⋀^{2} U_{3}^{\lor})^{\oplus 3} \oplus Sym^{2} U_{3}^{\lor} \oplus O (0, 2)$ 3 copies of wedge^3 Q_1^v ot wedge^2 U_2^v ot wedge^2 U_3^v + Sym^2 U_3^v + O(0,2)
ZL62 label: 709.666A7

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,6) \\ \det\mathcal{E} &\cong \mathcal{O}(3,5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	57
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	256
$\chi(X, \mathrm{T}_X)$	=	0

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	256
$\mathrm{c}_1^2\mathrm{c}_2$	=	160
$\mathrm{c}_2^2$	=	96
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	0

Hodge diamond

				1
			0		0
		0		2		0
	0		2		2		0
0		0		2		0		0
	0		2		2		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 + x₂	0	x₀	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 64	0	0	x₁	64
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 63	0	0	x₁ - x₂ + 6	0	57
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 7t + 7t^{2} + t^{3} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(442)
Coates et al. (2020)	P1 x B(3,4)
Coates–Kasprzyk–Prince (2015)	CKP(500)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 6t - 1 \right) \left( 6t + 1 \right) \left( 10t - 1 \right) \left( 10t + 1 \right) \left( 2t - 1 \right) ^{2} \left( 2t + 1 \right) ^{2} \left( 1433040t^{6} + 80728t^{4} - 2579t^{2} + 1 \right)D^{6} + 2 \left( 2t - 1 \right) \left( 2t + 1 \right) \left( 226993536000t^{12} - 6984049920t^{10} - 2174297216t^{8} + 67707232t^{6} - 106452t^{4} - 7441t^{2} + 2 \right)D^{5} + 4 \left( 3920797440000t^{14} - 584516290560t^{12} - 52264472064t^{10} + 4812821760t^{8} - 86750536t^{6} + 175464t^{4} - 4911t^{2} + 1 \right)D^{4} + 8 t ^{2} \left( 8460668160000t^{12} - 550314631680t^{10} - 112753881472t^{8} + 5245181728t^{6} - 97715360t^{4} - 329920t^{2} - 69 \right)D^{3} + 32 t ^{2} \left( 4769443728000t^{12} - 59034734400t^{10} - 53369502424t^{8} + 1881440232t^{6} - 26802560t^{4} - 94678t^{2} - 5 \right)D^{2} + 768 t ^{4} \left( 219040164000t^{10} + 4402781840t^{8} - 2035415446t^{6} + 67181120t^{4} - 620103t^{2} - 1737 \right)D + 4608 t ^{4} \left( 15046920000t^{10} + 593723200t^{8} - 120630700t^{6} + 4065994t^{4} - 22137t^{2} - 34 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 4, 8)

Show multiplicities

Sector dimension: 8
Spectral radius: $10$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-10$	1	1
$10$	1	1
$-6$	1	1
$6$	1	1
$-2$	2	2
$2$	2	2

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 6\right) \left(\lambda + 10\right) \left(\lambda - 10\right) \left(\lambda - 2\right)^{2} \left(\lambda - 6\right)\notag\end{equation}$$

Minimal polynomial