Family 1-243-55-a

← 1-256-57-a · 1-224-51-a →

A smooth cubic fourfold in $\mathbb P^5$.

This is a Fano fourfold of K3 type, appearing as case X3 in Bernardara–Fatighenti–Manivel–Tanturri.

Varieties in this family can be blown up to

2-102-27-a (from a member containing a Veronese surface)
2-115-30-b (from a member containing a plane)

ℙ⁵P5 · 50.A (preferred model)

Defining data

Ambient: ℙ⁵P5
Bundle: $E^{3 ω_{1}}$ E^(3w1)
Bundle: $O (3)$ O(3)
ZL62 label: 50.A

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	3
$\mathrm{h}^0(X, \omega_X^\vee)$	=	55
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	243
$\chi(X, \mathrm{T}_X)$	=	-20

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	243
$\mathrm{c}_1^2\mathrm{c}_2$	=	162
$\mathrm{c}_2^2$	=	108
$\mathrm{c}_1\mathrm{c}_3$	=	18
$\mathrm{c}_4$	=	27

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		1		21		1		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 + 20	0	20	x₀
$\mathrm{HH}^{ 3 }$ = x₀ + 50	0	0	x₀ - 20	70
$\mathrm{HH}^{ 4 }$ = 56	0	0	1	0	55
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(33)
Coates et al. (2020)	FI(3)
Coates–Kasprzyk–Prince (2015)	CKP(37)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 9t - 1 \right) \left( 81t^{2} + 9t + 1 \right)D^{4} + 4374 t ^{3}D^{3} + 9477 t ^{3}D^{2} + 8748 t ^{3}D + 2916 t ^{3} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ⁵P5

Show multiplicities

Sector dimension: 5
Spectral radius: $9$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-4.5 - 7.794229\mathrm{i}$	1	1
$-4.5 + 7.794229\mathrm{i}$	1	1
$9$	1	1
$0$	2	2

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda - 9\right) \left(\lambda^{2} + 9 \lambda + 81\right)\notag\end{equation}$$

Minimal polynomial