Family 1-96-27-a

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

ℙ⁶P6 · 600.6C (preferred model)

Defining data

Ambient: ℙ⁶P6
Bundle: $E^{2 ω_{1}} \oplus E^{3 ω_{1}}$ E^(2w1) + E^(3w1)
Bundle: $O (2) \oplus O (3)$ O(2) + O(3)
ZL62 label: 600.6C

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7) \\ \det\mathcal{E} &\cong \mathcal{O}(5) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	27
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	96
$\chi(X, \mathrm{T}_X)$	=	-55

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	96
$\mathrm{c}_1^2\mathrm{c}_2$	=	120
$\mathrm{c}_2^2$	=	150
$\mathrm{c}_1\mathrm{c}_3$	=	-24
$\mathrm{c}_4$	=	90

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		8		70		8		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
$\mathrm{HH}^{ 2 }$ = x₀ + 55	0	55	x₀
$\mathrm{HH}^{ 3 }$ = x₀ + 21	0	0	x₀	21
$\mathrm{HH}^{ 4 }$ = 55	0	0	28	0	27
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(630)
Coates et al. (2020)	V(4,6)
Coates–Kasprzyk–Prince (2015)	CKP(640)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 432t^{2} - 1 \right)D^{4} + 1728 t ^{2}D^{3} + 2544 t ^{2}D^{2} + 1632 t ^{2}D + 384 t ^{2} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁶P6

Show multiplicities

Sector dimension: 5
Spectral radius: $20.7846097$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-20.78461$	1	1
$20.78461$	1	1
$0$	3	3

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda^{2} - 432\right)\notag\end{equation}$$

Minimal polynomial