Family 2-272-60-e

← 2-272-60-d · 2-272-60-f →

Blow-up of $\mathbb P^4$ along a projected Veronese surface.

$\pi_1\colon X\to V_5$ is a $\mathbb P^1$-bundle over the del Pezzo threefold $V_5$ of degree $5$, a linear section of $\mathrm{Gr}(2,5)$ of codimension $3$, whose fibers are the lines of $\mathbb P^4$ parametrized by $V_5$ (the universal line).
$\pi_2\colon X\to\mathbb P^4$ is the blow-up of $\mathbb P^4$ (1-625-126-a) along a projected Veronese surface, the image of $\mathbb P^2$ embedded by conics in $\mathbb P^5$ and projected from a general point.

More details (pdf)

Varieties in this family can be blown down to

1-625-126-a

Fl(1, 2, 5) · 42.111 (preferred model)

Defining data

Ambient: Fl(1, 2, 5)
Bundle: $(E^{ω_{2}})^{\oplus 3}$ 3 copies of E^(w2)
Bundle: $O (0, 1)^{\oplus 3}$ 3 copies of O(0,1)
ZL62 label: 42.111

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,4) \\ \det\mathcal{E} &\cong \mathcal{O}(0,3) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	60
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	272
$\chi(X, \mathrm{T}_X)$	=	3

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	272
$\mathrm{c}_1^2\mathrm{c}_2$	=	164
$\mathrm{c}_2^2$	=	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 - 3	0	x₀ - 3	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 50	0	0	x₁ - 15	65
$\mathrm{HH}^{ 4 }$ = 60	0	0	0	0	60
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 55t + 160t^{2} + 55t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(109)
Coates et al. (2020)	No match
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( 4t + 1 \right) \left( 4199t^{7} - 4724t^{6} + 459t^{5} + 2053t^{4} + 351t^{3} + 9t^{2} + t - 1 \right) \left( 42533579123222t^{9} + 52777884924924t^{8} + 25992795080895t^{7} + 6237232989227t^{6} + 905946965649t^{5} + 124371344469t^{4} + 14985909786t^{3} + 802783321t^{2} + 2661253t + 156400 \right)D^{6} + \left( 15002273894026370952t^{17} + 8875089604020623610t^{16} - 5517834357840805612t^{15} - 3558689971972407387t^{14} + 2732590300103311630t^{13} + 3076891824885329312t^{12} + 1239291622186341189t^{11} + 293163107922091458t^{10} + 52915689819712130t^{9} + 8872070119930415t^{8} + 1290372498710968t^{7} + 135566884750060t^{6} + 10301734440100t^{5} + 849769120263t^{4} + 75707798375t^{3} + 3215348725t^{2} + 7045359t + 312800 \right)D^{5} + \left( 125018949116886424600t^{17} + 96538556064805186350t^{16} - 16341557521940099480t^{15} - 29908364790760406173t^{14} + 1509098924075388497t^{13} + 8806322888680357568t^{12} + 3817405595708486019t^{11} + 908465608693991107t^{10} + 171065022986004151t^{9} + 27835582112905997t^{8} + 3080257548028383t^{7} + 158873168194917t^{6} - 1852519452512t^{5} - 775981234815t^{4} - 68872257324t^{3} - 3136677952t^{2} - 9240671t - 156400 \right)D^{4} + t \left( 525079586290922983320t^{16} + 479974206375471748350t^{15} + 33565160265485109580t^{14} - 98126187624313042949t^{13} - 22493280875030149098t^{12} + 9914176098132363608t^{11} + 5694467848391672921t^{10} + 1475215503807017818t^{9} + 328848731620588180t^{8} + 62655524659449371t^{7} + 7743079665068858t^{6} + 514302960671346t^{5} + 15483068926298t^{4} + 248884007439t^{3} + 6159867741t^{2} + 85106415t + 70725 \right)D^{3} + 2 t ^{2} \left( 580087923902353010144t^{15} + 593237828402111497740t^{14} + 126184157761019811672t^{13} - 71504205356254143313t^{12} - 30874255016300837644t^{11} - 92962289423725949t^{10} + 1861602960589577445t^{9} + 607268967686715379t^{8} + 173708461292556189t^{7} + 37202131547005027t^{6} + 4552754175733933t^{5} + 279012453942476t^{4} + 7233459428864t^{3} + 150078057876t^{2} + 4117187792t + 33845504 \right)D^{2} + 8 t ^{2} \left( 157523875887276894996t^{15} + 173627824203722760855t^{14} + 52372024671528582234t^{13} - 10535346799632192779t^{12} - 7664693115615765878t^{11} - 1043200081354387670t^{10} + 30402571604900827t^{9} + 48047350268338246t^{8} + 23293886160405032t^{7} + 5838715140675026t^{6} + 717779646770501t^{5} + 41452287307353t^{4} + 939370655777t^{3} + 24680351970t^{2} + 610060294t + 3565920 \right)D + 8 t ^{2} \left( 64295459545827304080t^{15} + 74324541924677655900t^{14} + 26432337980605524840t^{13} - 1623750884871465012t^{12} - 2661497561792302348t^{11} - 546248664597196779t^{10} - 65416015921882504t^{9} - 1638239317357979t^{8} + 4704968354254757t^{7} + 1495020215376023t^{6} + 188333969529335t^{5} + 10511162165553t^{4} + 217032373031t^{3} + 7067143016t^{2} + 144561210t + 625600 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5)

Sector dimension: 8
Spectral radius: $9.04935242$
Diagonalizable at 1: Yes
Simple spectrum at 1: Yes

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$9.049352$	1	1
$-1.567957 - 6.746915\mathrm{i}$	1	1
$-1.567957 + 6.746915\mathrm{i}$	1	1
$-4$	1	1
$-3.992673$	1	1
$-2.553885$	1	1
$0.8165593 - 0.5307249\mathrm{i}$	1	1
$0.8165593 + 0.5307249\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda^{7} - \lambda^{6} - 9 \lambda^{5} - 351 \lambda^{4} - 2053 \lambda^{3} - 459 \lambda^{2} + 4724 \lambda - 4199\right)\notag\end{equation}$$

Minimal polynomial