Family 1-56-18-a

A codimension-three linear section of $\operatorname{OGr}(2,7)$.

Gr(2, 7) · 601.1116 (preferred model)

Defining data

Ambient: Gr(2, 7)
Bundle: $(E^{ω_{2}})^{\oplus 3} \oplus E^{2 ω_{1}}$ 3 copies of E^(w2) + E^(2w1)
Bundle: $O (1)^{\oplus 3} \oplus Sym^{2} U^{\lor}$ 3 copies of O(1) + Sym^2 U^v
ZL62 label: 601.1116

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	18
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	56
$\chi(X, \mathrm{T}_X)$	=	-33

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	56
$\mathrm{c}_1^2\mathrm{c}_2$	=	92
$\mathrm{c}_2^2$	=	152
$\mathrm{c}_1\mathrm{c}_3$	=	0
$\mathrm{c}_4$	=	48

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		3		38		3		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₂ + 33	0	x₀ + 33	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 1	0	0	x₁	1
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 15	0	0	x₁ - x₂ - 3	0	18
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 13t + 28t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	No match
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 8t + 1 \right) \left( 34t + 7 \right) ^{2} \left( 4t + 1 \right) ^{3} \left( 176t^{2} + 36t - 1 \right)D^{4} + 4 t \left( 34t + 7 \right) \left( 4t + 1 \right) ^{2} \left( 478720t^{4} + 292352t^{3} + 60864t^{2} + 4680t + 73 \right)D^{3} + 4 t \left( 4t + 1 \right) \left( 227870720t^{6} + 214601472t^{5} + 80083824t^{4} + 14810972t^{3} + 1366436t^{2} + 52729t + 329 \right)D^{2} + 2 t \left( 2604236800t^{7} + 2882625536t^{6} + 1306737792t^{5} + 309231424t^{4} + 40050680t^{3} + 2670444t^{2} + 71610t + 147 \right)D + 192 t ^{2} \left( 13021184t^{6} + 13730560t^{5} + 5882760t^{4} + 1300344t^{3} + 154330t^{2} + 9093t + 196 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 7)

Show multiplicities

Sector dimension: 6
Spectral radius: $40.3606798$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$40.36068$	1	1
$-8$	1	1
$-4.36068$	1	1
$-4$	3	3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda + 8\right) \left(\lambda^{2} - 36 \lambda - 176\right)\notag\end{equation}$$

Minimal polynomial