Family 8-80-21-a

← 6-172-40-a

Küchle’s fourfold of type (b9), parametrizing lines on an intersection of two quadrics in $\mathbb P^6$.

$X$, the Fano scheme of lines on a smooth complete intersection of two quadrics in $\mathbb P^6$, has Picard rank $8$ but no projections are described; its $64$ extremal contractions are all small and each contracts a plane to a point.

More details (pdf)

Gr(2, 7) · 601.66 (preferred model)

Defining data

Ambient: Gr(2, 7)
Bundle: $(E^{2 ω_{1}})^{\oplus 2}$ 2 copies of E^(2w1)
Bundle: $(Sym^{2} U^{\lor})^{\oplus 2}$ 2 copies of Sym^2 U^v
ZL62 label: 601.66

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	21
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	80
$\chi(X, \mathrm{T}_X)$	=	-4

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	80
$\mathrm{c}_1^2\mathrm{c}_2$	=	80
$\mathrm{c}_2^2$	=	160
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	48

Hodge diamond

				1
			0		0
		0		8		0
	0		0		0		0
0		0		30		0		0
	0		0		0		0
		0		8		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₂ + 4	0	x₀ + 4	x₂
$\mathrm{HH}^{ 3 }$ = x₁ + x₃	0	0	x₁	x₃
$\mathrm{HH}^{ 4 }$ = x₁ - x₂ + x₃ + 43	0	0	x₁ - x₂ + 10	x₃ + 12	21
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 16t + 46t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

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

$$\begin{equation} L_X=\left( 20t - 1 \right) \left( 12t + 1 \right) ^{2} \left( 36t - 5 \right) ^{2} \left( 4t - 1 \right) ^{3}D^{4} + 8 t \left( 12t + 1 \right) \left( 36t - 5 \right) \left( 4t - 1 \right) ^{2} \left( 43200t^{3} - 11088t^{2} + 836t + 1 \right)D^{3} + 4 t \left( 4t - 1 \right) \left( 522547200t^{6} - 256877568t^{5} + 42133248t^{4} - 1762944t^{3} - 158896t^{2} + 13356t + 5 \right)D^{2} + 32 \left( 36t - 5 \right) t ^{2} \left( 10368000t^{5} - 5750784t^{4} + 1053504t^{3} - 56976t^{2} - 2712t + 245 \right)D + 64 t ^{2} \left( 89579520t^{6} - 59470848t^{5} + 14637312t^{4} - 1531344t^{3} + 37512t^{2} + 4095t - 175 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 7)

Show multiplicities

Sector dimension: 6
Spectral radius: $20$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$20$	1	1
$-12$	2	2
$4$	3	3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 12\right)^{2} \left(\lambda - 20\right) \left(\lambda - 4\right)^{3}\notag\end{equation}$$

Minimal polynomial