Family 2-480-99-a

← 2-486-100-a · 2-433-90-a →

The projective bundle $\mathbb P_{Q^3}(\mathcal O\oplus\mathcal O(-1))$ over a smooth quadric threefold.

$\pi_1\colon X\to Q^3$ is the $\mathbb P^1$-bundle $\mathbb P_{Q^3}(\mathcal O\oplus\mathcal O(-1))$ over a smooth quadric threefold.
$\pi_2\colon X\to\mathbb P^5\times\mathbb P^4$ is an isomorphism onto its image.
$\pi_3\colon X\to\mathbb P^5\times\mathrm{Gr}(2,6)$ is an isomorphism onto its image.
$\pi_{23}\colon X\to C$ is the blow-up of the vertex of a quadric cone $C\subset\mathbb P^5$ of corank one (the cone over a smooth quadric threefold); the exceptional divisor is a quadric threefold with normal bundle $\mathcal O(-1)$.

More details (pdf)

Varieties in this family can be blown down to

A quadric cone in $\mathbb P^5$ of corank one

Fl(1, 2, 6) × ℙ⁴P4 · 4052.8Ew7 (preferred model)

Defining data

Ambient: Fl(1, 2, 6) × ℙ⁴P4
Bundle: $E^{ω_{5}} \oplus E^{ω_{5} + ω_{6}} \oplus E^{2 ω_{2}}$ E^(w5) + E^(w5 + w6) + E^(2w2)
Bundle: $⋀^{3} Q_{1, 1}^{\lor} \otimes U_{1, 2}^{\lor} \otimes U_{1, 3}^{\lor} \oplus ⋀^{3} Q_{1, 1}^{\lor} \otimes U_{1, 2}^{\lor} \otimes U_{1, 3}^{\lor} ⊠ O_{2} (1) \oplus O_{1} (0, 2)$ wedge^3 Q_1,1^v ot U_1,2^v ot U_1,3^v + wedge^3 Q_1,1^v ot U_1,2^v ot U_1,3^v box O_2(1) + O_1(0,2)
ZL62 label: 4052.8Ew7

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5;5) \\ \det\mathcal{E} &\cong \mathcal{O}(0,4;4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	99
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	480
$\chi(X, \mathrm{T}_X)$	=	16

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	480
$\mathrm{c}_1^2\mathrm{c}_2$	=	216
$\mathrm{c}_2^2$	=	96
$\mathrm{c}_1\mathrm{c}_3$	=	56
$\mathrm{c}_4$	=	8

Hodge diamond

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

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ - 16	0	x₀ - 16	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 76	0	0	x₁ - 80	156
$\mathrm{HH}^{ 4 }$ = 99	0	0	0	0	99
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 14t + 14t^{2} + t^{3} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(39)
Coates et al. (2020)	MW(13)
Coates–Kasprzyk–Prince (2015)	CKP(60)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t - 1 \right) \left( 2t + 1 \right) \left( 136t^{3} - 20t^{2} + 6t - 1 \right) \left( 136t^{3} + 20t^{2} + 6t + 1 \right) \left( 66683996t^{6} - 1058780t^{4} - 19394t^{2} + 15 \right)D^{6} + 4 \left( 27134518180352t^{14} - 3636686279168t^{12} - 76696829376t^{10} + 3834603264t^{8} - 107218310t^{6} + 2061400t^{4} + 29091t^{2} - 15 \right)D^{5} + 4 \left( 234343566103040t^{14} - 21804386465792t^{12} + 102596245216t^{10} + 15956069184t^{8} + 542705432t^{6} - 1694520t^{4} - 36762t^{2} + 15 \right)D^{4} + 32 t ^{2} \left( 126422186976640t^{12} - 8667906294144t^{10} + 129634375864t^{8} + 2690207848t^{6} + 17651600t^{4} - 204240t^{2} - 49 \right)D^{3} + 32 t ^{2} \left( 285066614292448t^{12} - 15534154687040t^{10} + 251203935368t^{8} - 291393828t^{6} - 1136928t^{4} - 258660t^{2} - 15 \right)D^{2} + 128 t ^{4} \left( 78551346664144t^{10} - 3663416463632t^{8} + 48257529676t^{6} - 820898270t^{4} - 4423658t^{2} - 37545 \right)D + 3072 t ^{4} \left( 1349017239080t^{10} - 57143052340t^{8} + 533807468t^{6} - 19445050t^{4} - 103234t^{2} - 345 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6) × ℙ⁴P4

Sector dimension: 8
Spectral radius: $6.269769$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-6.269769$	1	1
$6.269769$	1	1
$-0.1348845 - 4.655448\mathrm{i}$	1	1
$-0.1348845 + 4.655448\mathrm{i}$	1	1
$0.1348845 - 4.655448\mathrm{i}$	1	1
$0.1348845 + 4.655448\mathrm{i}$	1	1
$-2$	1	1
$2$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{3} + 6 \lambda^{2} + 20 \lambda + 136\right) \left(\lambda^{3} - 6 \lambda^{2} + 20 \lambda - 136\right)\notag\end{equation}$$

Minimal polynomial

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 0 & 0 & 8 & 0 & 0 \\ 2 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 & -4 & 0 & 0 & 8 \\ 0 & 4 & 2 & 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 2 & 0 & 0 & 2 & -4 & 0 \\ 0 & 0 & 0 & 4 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta