Family 1-42-15-a

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

Defining data

Ambient: Gr(2, 7)
Bundle: $(E^{ω_{2}})^{\oplus 6}$ 6 copies of E^(w2)
Bundle: $O (1)^{\oplus 6}$ 6 copies of O(1)
ZL62 label: 601.111111

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7) \\ \det\mathcal{E} &\cong \mathcal{O}(6) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	15
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	42
$\chi(X, \mathrm{T}_X)$	=	-42

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	42
$\mathrm{c}_1^2\mathrm{c}_2$	=	84
$\mathrm{c}_2^2$	=	171
$\mathrm{c}_1\mathrm{c}_3$	=	-14
$\mathrm{c}_4$	=	73

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		6		57		6		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = 0	0	0
$\mathrm{HH}^{ 2 }$ = x₀ + 42	0	42	x₀
$\mathrm{HH}^{ 3 }$ = x₀ + 6	0	0	x₀ + 6	0
$\mathrm{HH}^{ 4 }$ = 36	0	0	21	0	15
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 10t + 20t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(715)
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 14t + 3 \right) ^{2} \left( 5t + 1 \right) ^{3} \left( 2744t^{3} + 784t^{2} + 42t - 1 \right)D^{4} + 2 t \left( 14t + 3 \right) \left( 5t + 1 \right) ^{2} \left( 960400t^{4} + 543312t^{3} + 103292t^{2} + 7084t + 95 \right)D^{3} + t \left( 5t + 1 \right) \left( 470596000t^{6} + 416045280t^{5} + 144323424t^{4} + 24490788t^{3} + 2036174t^{2} + 68939t + 366 \right)D^{2} + t \left( 3361400000t^{7} + 3424786400t^{6} + 1418291280t^{5} + 303579696t^{4} + 35086464t^{3} + 2047680t^{2} + 46656t + 81 \right)D + 84 t ^{2} \left( 19208000t^{6} + 18796400t^{5} + 7402920t^{4} + 1485036t^{3} + 157114t^{2} + 8037t + 144 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 6)

Show multiplicities

Sector dimension: 6
Spectral radius: $56.6848428$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$56.68484$	1	1
$-9.688301$	1	1
$-5$	3	3
$-4.996542$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{3} - 42 \lambda^{2} - 784 \lambda - 2744\right)\notag\end{equation}$$

Minimal polynomial