Family 2-292-64-a

← 2-297-65-a · 2-288-63-a →

Blow-up of a smooth quadric fourfold along a del Pezzo surface of degree $4$.

$\pi_1\colon X\to Q^4$ is the blow-up of the smooth quadric fourfold (1-512-105-a) along a smooth quartic del Pezzo surface, cut out in $Q^4\subset\mathbb P^5$ by a hyperplane and a further quadric, i.e. a complete intersection of two quadrics in $\mathbb P^4$.
$\pi_2\colon X\to Y$ is the blow-up of the unique node $p$ of $Y\subset\mathbb P^6$, a complete intersection of two quadrics, one of which is a cone with vertex $p$; the exceptional divisor is a quadric threefold.

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Varieties in this family can be blown down to

1-512-105-a
A complete intersection of two quadrics in $\mathbb P^6$ with one node

Fl(1, 2, 7) · 602.57g (preferred model)

Defining data

Ambient: Fl(1, 2, 7)
Bundle: $E^{ω_{6}} \oplus E^{ω_{1} + ω_{2}} \oplus E^{2 ω_{2}}$ E^(w6) + E^(w1 + w2) + E^(2w2)
Bundle: $⋀^{4} Q_{1}^{\lor} \otimes U_{2}^{\lor} \otimes U_{3}^{\lor} \oplus O (1, 1) \oplus O (0, 2)$ wedge^4 Q_1^v ot U_2^v ot U_3^v + O(1,1) + O(0,2)
ZL62 label: 602.57g

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,6) \\ \det\mathcal{E} &\cong \mathcal{O}(1,4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	64
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	292
$\chi(X, \mathrm{T}_X)$	=	-3

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	292
$\mathrm{c}_1^2\mathrm{c}_2$	=	172
$\mathrm{c}_2^2$	=	98
$\mathrm{c}_1\mathrm{c}_3$	=	44
$\mathrm{c}_4$	=	14

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		8		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 + 3	0	x₀ + 3	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 54	0	0	x₁ - 25	79
$\mathrm{HH}^{ 4 }$ = 64	0	0	0	0	64
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 59t + 172t^{2} + 59t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(28)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(34)

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) ^{2} \left( 7936t^{5} + 1536t^{4} + 480t^{3} - 16t^{2} - t - 1 \right) \left( 2603769856t^{9} - 3624974336t^{8} - 5682914560t^{7} - 1958809088t^{6} - 125734192t^{5} - 2185376t^{4} - 3257386t^{3} - 471570t^{2} - 6078t - 73 \right)D^{6} + 2 \left( t + 1 \right) \left( 175639899406336t^{15} - 139928299634688t^{14} - 595638381084672t^{13} - 493849185845248t^{12} - 174876721793024t^{11} - 29809336997888t^{10} - 3502471186048t^{9} - 483530662272t^{8} - 108172911528t^{7} - 19777579648t^{6} - 1302563589t^{5} + 81154t^{4} - 7096878t^{3} - 921037t^{2} - 8825t - 73 \right)D^{5} + \left( 2376304521379840t^{16} - 606307856416768t^{15} - 10168719688794112t^{14} - 12485527552884736t^{13} - 6609972206792704t^{12} - 1708179744624640t^{11} - 217022785715968t^{10} - 25027740786432t^{9} - 5530811909104t^{8} - 869923321408t^{7} - 67070857422t^{6} - 4314358940t^{5} + 32886218t^{4} + 15516607t^{3} + 1801976t^{2} + 15606t + 73 \right)D^{4} + t \left( 8162089443000320t^{15} - 4910066271518720t^{14} - 34561999888908288t^{13} - 36671271713177600t^{12} - 17031398480654336t^{11} - 3793493447774208t^{10} - 418935663815424t^{9} - 61201090756096t^{8} - 17224186454384t^{7} - 2844171546288t^{6} - 226349176230t^{5} - 12576289642t^{4} - 226656512t^{3} - 2820796t^{2} - 3744t - 19 \right)D^{3} + t ^{2} \left( 14960386725904384t^{14} - 12858760685944832t^{13} - 61990402062942208t^{12} - 58606437841076224t^{11} - 24324814501314560t^{10} - 4765512881379328t^{9} - 527252652314112t^{8} - 109077885988608t^{7} - 29701703537280t^{6} - 4355647602544t^{5} - 280975896636t^{4} - 13130838244t^{3} - 269049308t^{2} - 3746300t - 3723 \right)D^{2} + 8 t ^{2} \left( 1725403717697536t^{14} - 1808920192745472t^{13} - 6986900570505216t^{12} - 6035456636231680t^{11} - 2283433984436224t^{10} - 402624599816704t^{9} - 49878395981568t^{8} - 13247180613312t^{7} - 3266124014664t^{6} - 426755619508t^{5} - 23422437228t^{4} - 932538120t^{3} - 21263153t^{2} - 288296t - 146 \right)D + 48 t ^{3} \left( 103317587886080t^{13} - 122114472935424t^{12} - 410227604324352t^{11} - 331089060446208t^{10} - 116527652384768t^{9} - 18960385258496t^{8} - 2747190245376t^{7} - 835307990912t^{6} - 188174965424t^{5} - 22267371080t^{4} - 1082778908t^{3} - 37556960t^{2} - 933021t - 11826 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 7)

Show multiplicities

Sector dimension: 8
Spectral radius: $8.32340409$
Diagonalizable at 1: No
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$8.323404$	1	1
$-2.859917 - 6.911588\mathrm{i}$	1	1
$-2.859917 + 6.911588\mathrm{i}$	1	1
$-1.801785 - 3.714168\mathrm{i}$	1	1
$-1.801785 + 3.714168\mathrm{i}$	1	1
$-1$	2	2
$0$	1	1

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda^{5} + \lambda^{4} + 16 \lambda^{3} - 480 \lambda^{2} - 1536 \lambda - 7936\right)\notag\end{equation}$$

Minimal polynomial