Family 2-431-90-a

← 2-432-90-d · 2-416-87-a →

Blow-up of a smooth quadric fourfold at a point.

$\pi_1\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a smooth quadric surface contained in a hyperplane; the composite with the inverse of $\pi_2$ is the projection of the smooth quadric fourfold (1-512-105-a) from a point.
$\pi_2\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) at a point.

More details (pdf)

Varieties in this family can be blown down to

Fl(1, 2, 6) · 52.4Z (preferred model)

Defining data

Ambient: Fl(1, 2, 6)
Bundle: $E^{ω_{5}} \oplus E^{ω_{1} + ω_{2}}$ E^(w5) + E^(w1 + w2)
Bundle: $⋀^{3} Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (1, 1)$ wedge^3 Q_1^v ot U_2^v ot U_3^v + O(1,1)
ZL62 label: 52.4Z

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5) \\ \det\mathcal{E} &\cong \mathcal{O}(1,2) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	90
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	431
$\chi(X, \mathrm{T}_X)$	=	11

$\omega_X^\vee \cong \mathcal{O}_X(1)$

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	431
$\mathrm{c}_1^2\mathrm{c}_2$	=	206
$\mathrm{c}_2^2$	=	94
$\mathrm{c}_1\mathrm{c}_3$	=	54
$\mathrm{c}_4$	=	9

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		3		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 - 11	0	x₀ - 11	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 71	0	0	x₁ - 64	135
$\mathrm{HH}^{ 4 }$ = 90	0	0	0	0	90
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 85t + 259t^{2} + 85t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(4)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(4)

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( t + 1 \right) \left( 30773t^{7} + 7500t^{6} + 5550t^{5} + 1132t^{4} - 27t^{3} - t^{2} - 2t - 1 \right) \left( 4036235283954t^{9} + 2642936629032t^{8} + 2354409402417t^{7} + 901371282609t^{6} + 78078590293t^{5} + 6557274217t^{4} + 1102940305t^{3} + 24407039t^{2} + 137615t + 10775 \right)D^{6} + \left( 2608348436255445282t^{17} + 4415424116861552706t^{16} + 4171904893209049191t^{15} + 3103401113259425199t^{14} + 1409808339481939205t^{13} + 446670580566182017t^{12} + 114134633923762352t^{11} + 17185574214620007t^{10} + 1684008553994647t^{9} + 240331949577180t^{8} + 46734362506269t^{7} + 8000520706377t^{6} + 573772951247t^{5} + 42491592019t^{4} + 5509620300t^{3} + 97021361t^{2} + 348195t + 21550 \right)D^{5} + \left( 21736236968795377350t^{17} + 33217972685807045670t^{16} + 31374902313403989039t^{15} + 22427846982578702166t^{14} + 9360704195009393746t^{13} + 2492636842716349176t^{12} + 548672390176020201t^{11} + 76685206031598656t^{10} + 7836629327748099t^{9} + 873212223093869t^{8} + 20812883889603t^{7} - 13353162326228t^{6} - 823470570226t^{5} - 36322090268t^{4} - 5290350179t^{3} - 89749882t^{2} - 266305t - 10775 \right)D^{4} + t \left( 91292195268940584870t^{16} + 127710631929860143110t^{15} + 121772583685143614925t^{14} + 84573828501805659321t^{13} + 32690309846186784643t^{12} + 7432649760576112901t^{11} + 1423273309549725658t^{10} + 191016444432727644t^{9} + 21230488531676384t^{8} + 2626040723715130t^{7} + 185242372970027t^{6} + 4658748110035t^{5} + 61768181657t^{4} + 1736151439t^{3} - 8176574t^{2} + 1460t + 1850 \right)D^{3} + 2 t ^{2} \left( 100856139535210550904t^{15} + 131111929228612266936t^{14} + 126812319530120668944t^{13} + 86040495484297870530t^{12} + 31008984510803801946t^{11} + 6186084913250684314t^{10} + 1057028211843368170t^{9} + 140650475268002988t^{8} + 15658608354760963t^{7} + 1909666413555733t^{6} + 119266502321295t^{5} + 3258705223403t^{4} + 57080182109t^{3} + 2326970355t^{2} + 3962429t - 2155 \right)D^{2} + 12 t ^{3} \left( 18258439053788116974t^{14} + 22412051633937705102t^{13} + 22001320212170213817t^{12} + 14644574082817570941t^{11} + 4965187136113398015t^{10} + 895317105778389617t^{9} + 141067085835406334t^{8} + 19080298435856880t^{7} + 2048921848783170t^{6} + 239449951990878t^{5} + 13176568844283t^{4} + 395824290859t^{3} + 8048948329t^{2} + 318882123t + 349110 \right)D + 144 t ^{4} \left( 621035341965582210t^{13} + 731896073534962530t^{12} + 727322996709380439t^{11} + 477249805562408415t^{10} + 154263503622282075t^{9} + 25866849743940009t^{8} + 3871799850560391t^{7} + 536971231241745t^{6} + 54687745901917t^{5} + 6046882265047t^{4} + 295462066728t^{3} + 9918083654t^{2} + 225651080t + 8275200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6)

Sector dimension: 8
Spectral radius: $6.32581249$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$6.325812$	1	1
$0.7131739 - 5.964656\mathrm{i}$	1	1
$0.7131739 + 5.964656\mathrm{i}$	1	1
$-4.765172 - 0.9897043\mathrm{i}$	1	1
$-4.765172 + 0.9897043\mathrm{i}$	1	1
$-0.1109078 - 2.383086\mathrm{i}$	1	1
$-0.1109078 + 2.383086\mathrm{i}$	1	1
$-1$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda^{7} + 2 \lambda^{6} + \lambda^{5} + 27 \lambda^{4} - 1132 \lambda^{3} - 5550 \lambda^{2} - 7500 \lambda - 30773\right)\notag\end{equation}$$

Minimal polynomial