Family 1-33-13-a

A codimension-two linear section of the zero locus of $\mathcal U^\vee(1)$ on $\operatorname{Gr}(2,6)$, Küchle’s fourfold of type (b5).

Gr(2, 6) · 51.11Z (preferred model)

Defining data

Ambient: Gr(2, 6)
Bundle: $(E^{ω_{2}})^{\oplus 2} \oplus E^{ω_{1} + ω_{2}}$ 2 copies of E^(w2) + E^(w1 + w2)
Bundle: $O (1)^{\oplus 2} \oplus U^{\lor} (1)$ 2 copies of O(1) + U^v(1)
ZL62 label: 51.11Z

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	13
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	33
$\chi(X, \mathrm{T}_X)$	=	-48

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	33
$\mathrm{c}_1^2\mathrm{c}_2$	=	78
$\mathrm{c}_2^2$	=	185
$\mathrm{c}_1\mathrm{c}_3$	=	-24
$\mathrm{c}_4$	=	90

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		8		70		8		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 + 48	0	48	x₀
$\mathrm{HH}^{ 3 }$ = x₀ + 2	0	0	x₀ + 2	0
$\mathrm{HH}^{ 4 }$ = 43	0	0	30	0	13
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 8t + 15t^{2} + 8t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(730)
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 84t + 11 \right) ^{2} \left( 7t + 1 \right) ^{3} \left( 4887t^{3} + 1188t^{2} + 63t - 1 \right)D^{4} + 6 t \left( 84t + 11 \right) \left( 7t + 1 \right) ^{2} \left( 4789260t^{4} + 1957662t^{3} + 276525t^{2} + 14788t + 182 \right)D^{3} + t \left( 7t + 1 \right) \left( 59137782480t^{6} + 35797617576t^{5} + 8618367303t^{4} + 1034016732t^{3} + 62484336t^{2} + 1613464t + 7700 \right)D^{2} + t \left( 591377824800t^{7} + 411395772312t^{6} + 117575488926t^{5} + 17614041858t^{4} + 1452921204t^{3} + 62330565t^{2} + 1099296t + 1694 \right)D + 6 t ^{2} \left( 47310225984t^{6} + 31376786616t^{5} + 8478247932t^{4} + 1186308228t^{3} + 89634132t^{2} + 3397911t + 48400 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 6)

Show multiplicities

Sector dimension: 6
Spectral radius: $78.8521567$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$78.85216$	1	1
$-8.845846$	1	1
$-7.00631$	1	1
$-7$	3	3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 7\right)^{3} \left(\lambda^{3} - 63 \lambda^{2} - 1188 \lambda - 4887\right)\notag\end{equation}$$

Minimal polynomial