Family 1-324-70-a

← 1-405-85-a · 1-288-63-a →

A smooth complete intersection of two quadrics in $\mathbb P^6$.

Varieties in this family can be blown up to

2-116-30-c (from a member containing a plane)
2-147-36-a
2-244-55-a

ℙ⁶P6 · 600.66 (preferred model)

Defining data

Ambient: ℙ⁶P6
Bundle: $(E^{2 ω_{1}})^{\oplus 2}$ 2 copies of E^(2w1)
Bundle: $O (2)^{\oplus 2}$ 2 copies of O(2)
ZL62 label: 600.66

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(7) \\ \det\mathcal{E} &\cong \mathcal{O}(4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	3
$\mathrm{h}^0(X, \omega_X^\vee)$	=	70
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	324
$\chi(X, \mathrm{T}_X)$	=	-4

$\omega_X^\vee \cong \mathcal{O}_X(3)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	324
$\mathrm{c}_1^2\mathrm{c}_2$	=	180
$\mathrm{c}_2^2$	=	100
$\mathrm{c}_1\mathrm{c}_3$	=	36
$\mathrm{c}_4$	=	12

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		0		8		0		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
$\mathrm{HH}^{ 2 }$ = 39	0	4	35
$\mathrm{HH}^{ 3 }$ = 98	0	0	0	98
$\mathrm{HH}^{ 4 }$ = 70	0	0	0	0	70
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(27)
Coates et al. (2020)	FI(4)
Coates–Kasprzyk–Prince (2015)	CKP(33)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 432t^{3} - 1 \right)D^{4} + 2592 t ^{3}D^{3} + 5724 t ^{3}D^{2} + 5508 t ^{3}D + 1944 t ^{3} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁶P6

Show multiplicities

Sector dimension: 5
Spectral radius: $7.5595263$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-3.779763 - 6.546742\mathrm{i}$	1	1
$-3.779763 + 6.546742\mathrm{i}$	1	1
$7.559526$	1	1
$0$	2	2

Characteristic polynomial

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

Minimal polynomial