Family 1-66-20-a

This is a Fano fourfold of K3 type, appearing as case Z66 in Bernardara–Fatighenti–Manivel–Tanturri.

Gr(3, 7) · 603.321 (preferred model)

Defining data

Ambient: Gr(3, 7)
Bundle: $E^{ω_{2}} \oplus E^{ω_{3}} \oplus E^{ω_{4}}$ E^(w2) + E^(w3) + E^(w4)
Bundle: $⋀^{2} U^{\lor} \oplus O (1) \oplus Q^{\lor} (1)$ wedge^2 U^v + O(1) + Q^v(1)
ZL62 label: 603.321

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	20
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	66
$\chi(X, \mathrm{T}_X)$	=	-25

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	66
$\mathrm{c}_1^2\mathrm{c}_2$	=	96
$\mathrm{c}_2^2$	=	140
$\mathrm{c}_1\mathrm{c}_3$	=	12
$\mathrm{c}_4$	=	30

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		1		24		1		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 + 25	0	x₀ + 25	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + 1	0	0	x₁	1
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + 5	0	0	x₁ - x₂ - 15	0	20
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 15t + 34t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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

$$\begin{equation} L_X=\left( 5t + 1 \right) \left( 4t + 1 \right) ^{2} \left( 36t + 11 \right) ^{2} \left( 432t^{3} + 216t^{2} + 27t - 1 \right)D^{4} + 4 t \left( 4t + 1 \right) \left( 36t + 11 \right) \left( 777600t^{5} + 837216t^{4} + 344088t^{3} + 65322t^{2} + 5241t + 95 \right)D^{3} + t \left( 1567641600t^{7} + 2358429696t^{6} + 1464887808t^{5} + 481005504t^{4} + 88223520t^{3} + 8641292t^{2} + 371087t + 2695 \right)D^{2} + t \left( 2239488000t^{7} + 3174598656t^{6} + 1834161408t^{5} + 550689984t^{4} + 90155856t^{3} + 7576404t^{2} + 255266t + 605 \right)D + 72 t ^{2} \left( 14929920t^{6} + 20362752t^{5} + 11205696t^{4} + 3158944t^{3} + 475416t^{2} + 35409t + 968 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(3, 7)

Show multiplicities

Sector dimension: 6
Spectral radius: $33.7741424$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$33.77414$	1	1
$-5$	1	1
$-4$	2	2
$-3.387071 - 1.148303\mathrm{i}$	1	1
$-3.387071 + 1.148303\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right) \left(\lambda^{3} - 27 \lambda^{2} - 216 \lambda - 432\right)\notag\end{equation}$$

Minimal polynomial