Family 1-12-8-a

A smooth complete intersection of two quadrics and a cubic in $\mathbb P^7$.

ℙ⁷P7 · 700.77E (preferred model)

Defining data

Ambient: ℙ⁷P7
Bundle: $(E^{2 ω_{1}})^{\oplus 2} \oplus E^{3 ω_{1}}$ 2 copies of E^(2w1) + E^(3w1)
Bundle: $O (2)^{\oplus 2} \oplus O (3)$ 2 copies of O(2) + O(3)
ZL62 label: 700.77E

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(8) \\ \det\mathcal{E} &\cong \mathcal{O}(7) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	8
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	12
$\chi(X, \mathrm{T}_X)$	=	-108

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	12
$\mathrm{c}_1^2\mathrm{c}_2$	=	60
$\mathrm{c}_2^2$	=	300
$\mathrm{c}_1\mathrm{c}_3$	=	-84
$\mathrm{c}_4$	=	324

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		42		236		42		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
$\mathrm{HH}^{ 2 }$ = 108	0	108	0
$\mathrm{HH}^{ 3 }$ = 0	0	0	0	0
$\mathrm{HH}^{ 4 }$ = 205	0	0	197	0	8
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 10			0	10	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(746)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(735)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 408t - 1 \right) \left( 24t + 1 \right) ^{4}D^{4} + 48 t \left( 2040t + 13 \right) \left( 24t + 1 \right) ^{3}D^{3} + 36 t \left( 24t + 1 \right) ^{2} \left( 228480t^{2} + 5616t + 11 \right)D^{2} + 12 t \left( 24t + 1 \right) \left( 23500800t^{3} + 1181952t^{2} + 12888t + 7 \right)D + 1152 t ^{2} \left( 2820096t^{3} + 231552t^{2} + 5652t + 37 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁷P7

Show multiplicities

Sector dimension: 5
Spectral radius: $408$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$408$	1	1
$-24$	4	4

Characteristic polynomial

$$\begin{equation}\left(\lambda + 24\right)^{4} \left(\lambda - 408\right)\notag\end{equation}$$

Minimal polynomial