Family 1-288-63-a

A smooth hyperplane section of the adjoint $G_2$ fivefold.

Varieties in this family can be blown up to

Gr(2, 7) · 601.21 (preferred model)

Defining data

Ambient: Gr(2, 7)
Bundle: $E^{ω_{2}} \oplus E^{ω_{3}}$ E^(w2) + E^(w3)
Bundle: $O (1) \oplus Q^{\lor} (1)$ O(1) + Q^v(1)
ZL62 label: 601.21

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7) \\ \det\mathcal{E} &\cong \mathcal{O}(5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	63
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	288
$\chi(X, \mathrm{T}_X)$	=	1

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	288
$\mathrm{c}_1^2\mathrm{c}_2$	=	168
$\mathrm{c}_2^2$	=	98
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	6

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		0		2		0		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 + 19	0	x₀ - 1	20
$\mathrm{HH}^{ 3 }$ = 77	0	0	0	77
$\mathrm{HH}^{ 4 }$ = 63	0	0	0	0	63
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 8t + 8t^{2} + t^{3} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	No match
Coates et al. (2020)	V(4,18)
Coates–Kasprzyk–Prince (2015)	No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 432t^{4} + 72t^{2} - 1 \right)D^{4} + 288 t ^{2} \left( 12t^{2} + 1 \right)D^{3} + 24 t ^{2} \left( 414t^{2} + 19 \right)D^{2} + 336 t ^{2} \left( 36t^{2} + 1 \right)D + 96 t ^{2} \left( 54t^{2} + 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 7)

Sector dimension: 5
Spectral radius: $8.80733895$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-8.807339$	1	1
$8.807339$	1	1
$0 - 2.359919\mathrm{i}$	1	1
$0 + 2.359919\mathrm{i}$	1	1
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{4} - 72 \lambda^{2} - 432\right)\notag\end{equation}$$

Minimal polynomial