Family 2-512-105-a

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

Defining data

Ambient: ℙ²P2 × Fl(2, 3, 7)
Bundle: $(E^{ω_{8}})^{\oplus 2} \oplus E^{ω_{2} + ω_{3}}$ 2 copies of E^(w8) + E^(w2 + w3)
Bundle: $(⋀^{3} Q_{2, 1}^{\lor} \otimes U_{2, 2}^{\lor} \otimes ⋀^{2} U_{2, 3}^{\lor})^{\oplus 2} \oplus Q_{1} ⊠ U_{2, 3}^{\lor}$ 2 copies of wedge^3 Q_2,1^v ot U_2,2^v ot wedge^2 U_2,3^v + Q_1 box U_2,3^v
ZL62 label: 20605.77w7

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	105
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	512
$\chi(X, \mathrm{T}_X)$	=	18

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	512
$\mathrm{c}_1^2\mathrm{c}_2$	=	224
$\mathrm{c}_2^2$	=	96
$\mathrm{c}_1\mathrm{c}_3$	=	56
$\mathrm{c}_4$	=	8

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		2		0		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 - 18	0	x₀ - 18	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 80	0	0	x₁ - 90	170
$\mathrm{HH}^{ 4 }$ = 105	0	0	0	0	105
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 15t + 15t^{2} + t^{3} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(36)
Coates et al. (2020)	P1 x P3
Coates–Kasprzyk–Prince (2015)	CKP(51)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t - 1 \right) \left( 2t + 1 \right) \left( 6t - 1 \right) \left( 6t + 1 \right) \left( 20t^{2} - 4t + 1 \right) \left( 20t^{2} + 4t + 1 \right) \left( 127920t^{6} - 45016t^{4} - 293t^{2} - 1 \right)D^{6} + 2 \left( 81050112000t^{14} - 42347458560t^{12} + 4138403072t^{10} + 109225472t^{8} + 95280t^{6} + 187248t^{4} + 911t^{2} + 2 \right)D^{5} + 4 \left( 349989120000t^{14} - 177028331520t^{12} + 12857605632t^{10} + 227619584t^{8} + 1578760t^{6} - 171496t^{4} - 769t^{2} - 1 \right)D^{4} + 8 t ^{2} \left( 755239680000t^{12} - 383806594560t^{10} + 19129972096t^{8} + 278237408t^{6} + 1853984t^{4} + 10304t^{2} + 13 \right)D^{3} + 32 t ^{2} \left( 425743344000t^{12} - 221217604800t^{10} + 7159648792t^{8} + 109530520t^{6} + 1224240t^{4} + 4582t^{2} + 1 \right)D^{2} + 768 t ^{4} \left( 19552572000t^{10} - 10422296720t^{8} + 211812198t^{6} + 4163952t^{4} + 55827t^{2} + 173 \right)D + 23040 t ^{4} \left( 268632000t^{10} - 146189120t^{8} + 1864348t^{6} + 52366t^{4} + 729t^{2} + 2 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ³P3 × ℙ¹P1

Sector dimension: 8
Spectral radius: $6$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-6$	1	1
$6$	1	1
$-2 - 4\mathrm{i}$	1	1
$-2 + 4\mathrm{i}$	1	1
$2 - 4\mathrm{i}$	1	1
$2 + 4\mathrm{i}$	1	1
$-2$	1	1
$2$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 6\right) \left(\lambda - 2\right) \left(\lambda - 6\right) \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{2} - 4 \lambda + 20\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 0 & 0 & 0 & 4 & 0 \\ 4 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 4 \\ 0 & 2 & 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 4 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 4 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta