Family 1-405-85-a

The del Pezzo fourfold of degree $5$, a codimension-two linear section of $\operatorname{Gr}(2,5)$.

Gr(2, 5) · 41.11 (preferred model)

Defining data

Ambient: Gr(2, 5)
Bundle: $(E^{ω_{2}})^{\oplus 2}$ 2 copies of E^(w2)
Bundle: $O (1)^{\oplus 2}$ 2 copies of O(1)
ZL62 label: 41.11

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5) \\ \det\mathcal{E} &\cong \mathcal{O}(2) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	3
$\mathrm{h}^0(X, \omega_X^\vee)$	=	85
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	405
$\chi(X, \mathrm{T}_X)$	=	8

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	405
$\mathrm{c}_1^2\mathrm{c}_2$	=	198
$\mathrm{c}_2^2$	=	97
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	6

Hodge diamond

				1
			0		0
		0		1		0
	0		0		0		0
0		0		2		0		0
	0		0		0		0
		0		1		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = 8	0	8
$\mathrm{HH}^{ 2 }$ = 56	0	0	56
$\mathrm{HH}^{ 3 }$ = 128	0	0	0	128
$\mathrm{HH}^{ 4 }$ = 85	0	0	0	0	85
$\mathrm{HH}^{ 5 }$ = 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

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

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

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(24)
Coates et al. (2020)	FI(5)
Coates–Kasprzyk–Prince (2015)	No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 729t^{6} + 297t^{3} - 1 \right)D^{4} + 162 t ^{3} \left( 54t^{3} + 11 \right)D^{3} + 27 t ^{3} \left( 1323t^{3} + 148 \right)D^{2} + 81 t ^{3} \left( 702t^{3} + 49 \right)D + 1458 t ^{3} \left( 20t^{3} + 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 5)

Sector dimension: 6
Spectral radius: $6.69012124$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-3.345061 - 5.793815\mathrm{i}$	1	1
$-3.345061 + 5.793815\mathrm{i}$	1	1
$6.690121$	1	1
$-1.345267$	1	1
$0.6726335 - 1.165035\mathrm{i}$	1	1
$0.6726335 + 1.165035\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\lambda^{6} - 297 \lambda^{3} - 729\notag\end{equation}$$

Minimal polynomial