Family 1-16-9-a

A smooth complete intersection of four quadrics in $\mathbb P^8$.

ℙ⁸P8 · 800.8888 (preferred model)

Defining data

Ambient: ℙ⁸P8
Bundle: $(E^{2 ω_{1}})^{\oplus 4}$ 4 copies of E^(2w1)
Bundle: $O (2)^{\oplus 4}$ 4 copies of O(2)
ZL62 label: 800.8888

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(9) \\ \det\mathcal{E} &\cong \mathcal{O}(8) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	9
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	16
$\chi(X, \mathrm{T}_X)$	=	-84

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	16
$\mathrm{c}_1^2\mathrm{c}_2$	=	64
$\mathrm{c}_2^2$	=	256
$\mathrm{c}_1\mathrm{c}_3$	=	-64
$\mathrm{c}_4$	=	224

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		27		166		27		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
$\mathrm{HH}^{ 2 }$ = 84	0	84	0
$\mathrm{HH}^{ 3 }$ = 0	0	0	0	0
$\mathrm{HH}^{ 4 }$ = 135	0	0	126	0	9
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 4			0	4	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

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

$\displaystyle\frac{ 1 + 4t + 6t^{2} + 4t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(741)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(731)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 240t - 1 \right) \left( 16t + 1 \right) ^{4}D^{4} + 32 t \left( 1200t + 11 \right) \left( 16t + 1 \right) ^{3}D^{3} + 32 t \left( 16t + 1 \right) ^{2} \left( 67200t^{2} + 2456t + 7 \right)D^{2} + 16 t \left( 16t + 1 \right) \left( 3072000t^{3} + 230912t^{2} + 3760t + 3 \right)D + 256 t ^{2} \left( 1474560t^{3} + 181248t^{2} + 6624t + 65 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁸P8

Show multiplicities

Sector dimension: 5
Spectral radius: $240$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$240$	1	1
$-16$	4	4

Characteristic polynomial

$$\begin{equation}\left(\lambda + 16\right)^{4} \left(\lambda - 240\right)\notag\end{equation}$$

Minimal polynomial