Family 1-61-19-a

A codimension-two linear section of the bisymplectic Grassmannian $\operatorname{I}_2\operatorname{Gr}(3,7)$.

Gr(3, 7) · 603.2211 (preferred model)

Defining data

Ambient: Gr(3, 7)
Bundle: $(E^{ω_{2}})^{\oplus 2} \oplus (E^{ω_{3}})^{\oplus 2}$ 2 copies of E^(w2) + 2 copies of E^(w3)
Bundle: $(⋀^{2} U^{\lor})^{\oplus 2} \oplus O (1)^{\oplus 2}$ 2 copies of wedge^2 U^v + 2 copies of O(1)
ZL62 label: 603.2211

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	19
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	61
$\chi(X, \mathrm{T}_X)$	=	-28

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	61
$\mathrm{c}_1^2\mathrm{c}_2$	=	94
$\mathrm{c}_2^2$	=	145
$\mathrm{c}_1\mathrm{c}_3$	=	8
$\mathrm{c}_4$	=	38

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		2		30		2		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 + x₁ + 28	0	x₀ + 28	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 14	0	0	x₁ + 14	0
$\mathrm{HH}^{ 4 }$ = 23	0	0	4	0	19
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 14t + 31t^{2} + 14t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 224t + 61 \right) ^{2} \left( 5t + 1 \right) ^{3} \left( 293t^{3} + 195t^{2} + 32t - 1 \right)D^{4} + 2 t \left( 224t + 61 \right) \left( 5t + 1 \right) ^{2} \left( 1640800t^{4} + 1541054t^{3} + 502194t^{2} + 60590t + 1261 \right)D^{3} + t \left( 5t + 1 \right) \left( 12863872000t^{6} + 16332532160t^{5} + 8226951733t^{4} + 2053006369t^{3} + 254400487t^{2} + 12966797t + 99247 \right)D^{2} + t \left( 91884800000t^{7} + 126172580800t^{6} + 71012714910t^{5} + 20886378172t^{4} + 3367189424t^{3} + 279910761t^{2} + 9384240t + 22326 \right)D + 2 t ^{2} \left( 22052352000t^{6} + 28967265600t^{5} + 15476500980t^{4} + 4274001704t^{3} + 636131976t^{2} + 47310441t + 1309792 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(3, 7)

Show multiplicities

Sector dimension: 6
Spectral radius: $37.4203168$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$37.42032$	1	1
$-5$	3	3
$-2.710158 - 0.6964285\mathrm{i}$	1	1
$-2.710158 + 0.6964285\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{3} - 32 \lambda^{2} - 195 \lambda - 293\right)\notag\end{equation}$$

Minimal polynomial