Family 1-8-7-a

A smooth complete intersection of a quadric and a quartic in $\mathbb P^6$.

ℙ⁶P6 · 600.6I (preferred model)

Defining data

Ambient: ℙ⁶P6
Bundle: $E^{2 ω_{1}} \oplus E^{4 ω_{1}}$ E^(2w1) + E^(4w1)
Bundle: $O (2) \oplus O (4)$ O(2) + O(4)
ZL62 label: 600.6I

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
$\mathrm{h}^0(X, \omega_X^\vee)$	=	7
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	8
$\chi(X, \mathrm{T}_X)$	=	-160

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	8
$\mathrm{c}_1^2\mathrm{c}_2$	=	56
$\mathrm{c}_2^2$	=	392
$\mathrm{c}_1\mathrm{c}_3$	=	-120
$\mathrm{c}_4$	=	552

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		77		394		77		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
$\mathrm{HH}^{ 2 }$ = 160	0	160	0
$\mathrm{HH}^{ 3 }$ = 0	0	0	0	0
$\mathrm{HH}^{ 4 }$ = 363	0	0	356	0	7
$\mathrm{HH}^{ 5 }$ = 0		0	0	0	0
$\mathrm{HH}^{ 6 }$ = 28			0	28	0
$\mathrm{HH}^{ 7 }$ = 0				0	0
$\mathrm{HH}^{ 8 }$ = 0					0

Hilbert series

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(748)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(737)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 976t - 1 \right) \left( 48t + 1 \right) ^{4}D^{4} + 32 t \left( 14640t + 49 \right) \left( 48t + 1 \right) ^{3}D^{3} + 32 t \left( 48t + 1 \right) ^{2} \left( 2459520t^{2} + 30552t + 31 \right)D^{2} + 16 t \left( 48t + 1 \right) \left( 337305600t^{3} + 8520192t^{2} + 46704t + 13 \right)D + 2304 t ^{2} \left( 53968896t^{3} + 2221056t^{2} + 27168t + 89 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁶P6

Show multiplicities

Sector dimension: 5
Spectral radius: $976$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$976$	1	1
$-48$	4	4

Characteristic polynomial

$$\begin{equation}\left(\lambda + 48\right)^{4} \left(\lambda - 976\right)\notag\end{equation}$$

Minimal polynomial