Family 2-416-87-a

← 2-431-90-a · 2-384-81-a →

Blow-up of a smooth quadric fourfold along a line.

$\pi_1\colon X\to\Gamma\cong\mathbb P^3$, with $\Gamma\subset\mathrm{Gr}(3,6)\times\mathbb P^3$ the graph of an isomorphism, is the resolved linear projection of the smooth quadric fourfold $Q^4$ (1-512-105-a) from a line $\ell$ on it; its general fiber is $\mathbb P^1$, and exactly two fibers, over the points corresponding to the two planes of $Q^4$ through $\ell$, are planes $\mathbb P^2$.
$\pi_2\colon X\to\mathbb P^5\times\mathbb P^3$ is an isomorphism onto its image.
$\pi_3\colon X\to\mathbb P^5\times\mathrm{Gr}(3,6)$ is an isomorphism onto its image.
$\pi_{23}\colon X\to Q^4\subset\mathbb P^5$ is the blow-up of the smooth quadric fourfold $Q^4$ (1-512-105-a) along a line $\ell$.

More details (pdf)

Varieties in this family can be blown down to

1-512-105-a

Fl(1, 3, 6) × ℙ³P3 · 3054.77wJw6 (preferred model)

Defining data

Ambient: Fl(1, 3, 6) × ℙ³P3
Bundle: $(E^{ω_{5}})^{\oplus 2} \oplus E^{ω_{5} + ω_{6}} \oplus E^{ω_{1} + ω_{3}}$ 2 copies of E^(w5) + E^(w5 + w6) + E^(w1 + w3)
Bundle: $(⋀^{2} Q_{1, 1}^{\lor} \otimes ⋀^{2} U_{1, 2}^{\lor} \otimes U_{1, 3}^{\lor})^{\oplus 2} \oplus ⋀^{2} Q_{1, 1}^{\lor} \otimes ⋀^{2} U_{1, 2}^{\lor} \otimes U_{1, 3}^{\lor} ⊠ O_{2} (1) \oplus O_{1} (1, 1)$ 2 copies of wedge^2 Q_1,1^v ot wedge^2 U_1,2^v ot U_1,3^v + wedge^2 Q_1,1^v ot wedge^2 U_1,2^v ot U_1,3^v box O_2(1) + O_1(1,1)
ZL62 label: 3054.77wJw6

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,5;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,4;3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	2
$\mathrm{h}^0(X, \omega_X^\vee)$	=	87
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	416
$\chi(X, \mathrm{T}_X)$	=	10

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	416
$\mathrm{c}_1^2\mathrm{c}_2$	=	200
$\mathrm{c}_2^2$	=	98
$\mathrm{c}_1\mathrm{c}_3$	=	52
$\mathrm{c}_4$	=	10

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		4		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 - 10	0	x₀ - 10	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 68	0	0	x₁ - 62	130
$\mathrm{HH}^{ 4 }$ = 87	0	0	0	0	87
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 12t + 12t^{2} + t^{3} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(40)
Coates et al. (2020)	MW(12)
Coates–Kasprzyk–Prince (2015)	CKP(61)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 16t^{4} + 44t^{2} - 1 \right) \left( 432t^{4} + 36t^{2} + 1 \right) \left( 314924112t^{8} - 117964512t^{6} + 14238144t^{4} + 164850t^{2} + 221 \right)D^{6} + 4 \left( 10883777310720t^{16} + 14017882540032t^{14} - 7127443839744t^{12} + 913572411264t^{10} + 49215528432t^{8} + 208091040t^{6} + 29910072t^{4} + 249043t^{2} + 221 \right)D^{5} + 4 \left( 83805085292544t^{16} + 42638961696768t^{14} - 38573795723328t^{12} + 6607519967520t^{10} + 197257285008t^{8} + 2651761872t^{6} - 52981980t^{4} - 386208t^{2} - 221 \right)D^{4} + 16 t ^{2} \left( 79451574368256t^{14} + 656487804672t^{12} - 24209536131072t^{10} + 5352527374416t^{8} + 119397810240t^{6} + 1063433832t^{4} + 1841528t^{2} + 1513 \right)D^{3} + 32 t ^{2} \left( 78431220245376t^{14} - 24146078481216t^{12} - 14886119891520t^{10} + 4353594087744t^{8} + 89900843112t^{6} + 822085824t^{4} + 1734954t^{2} + 221 \right)D^{2} + 2304 t ^{4} \left( 1061924105664t^{12} - 538846899984t^{10} - 114223084416t^{8} + 48172072716t^{6} + 1084223724t^{4} + 9219514t^{2} + 22737 \right)D + 27648 t ^{4} \left( 33067031760t^{12} - 20911507560t^{10} - 1683208512t^{8} + 1243973394t^{6} + 32605206t^{4} + 248744t^{2} + 663 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 6) × ℙ³P3

Sector dimension: 9
Spectral radius: $6.66038135$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$-6.660381$	1	1
$6.660381$	1	1
$-1.17996 - 4.403669\mathrm{i}$	1	1
$-1.17996 + 4.403669\mathrm{i}$	1	1
$1.17996 - 4.403669\mathrm{i}$	1	1
$1.17996 + 4.403669\mathrm{i}$	1	1
$0 - 0.6005662\mathrm{i}$	1	1
$0 + 0.6005662\mathrm{i}$	1	1
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{4} + 36 \lambda^{2} + 432\right) \left(\lambda^{4} - 44 \lambda^{2} - 16\right)\notag\end{equation}$$

Minimal polynomial