Family 2-146-36-d

← 2-146-36-c · 2-144-36-a →

Blow-up of the genus-seven Mukai fourfold along a quadric surface.

$\pi_1\colon X\to Z$ is the blow-up $\mathrm{Bl}_\Sigma Z$ of the Fano fourfold $Z\subset\mathrm{Gr}(2,5)$ of type 1-192-45-a along a smooth quadric surface $\Sigma\cong\mathbb P^1\times\mathbb P^1$, contained in a linearly embedded $\mathbb P^3\subset\mathrm{Gr}(2,5)$ (the lines of $\mathbb P^4$ through a point).
$\pi_2\colon X\to\mathbb Q^4$ is the blow-up $\mathrm{Bl}_S\mathbb Q^4$ of the smooth quadric fourfold $\mathbb Q^4=\mathrm{Gr}(2,4)$ (1-512-105-a) along a smooth rational surface $S$ of degree $8$, isomorphic to the blow-up of $\mathbb P^1\times\mathbb P^1$ at ten points and embedded by the curves of bidegree $(3,3)$ through them.

More details (pdf)

Varieties in this family can be blown down to

Gr(2, 4) × Gr(2, 5) · 3141.wCw2 (preferred model)

Defining data

Ambient: Gr(2, 4) × Gr(2, 5)
Bundle: $E^{ω_{1} + ω_{4}} \oplus E^{ω_{3} + ω_{5}}$ E^(w1 + w4) + E^(w3 + w5)
Bundle: $U_{1}^{\lor} ⊠ U_{2}^{\lor} \oplus Q_{1} ⊠ O_{2} (1)$ U_1^v box U_2^v + Q_1 box O_2(1)
ZL62 label: 3141.wCw2

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	36
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	146
$\chi(X, \mathrm{T}_X)$	=	-15

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	146
$\mathrm{c}_1^2\mathrm{c}_2$	=	128
$\mathrm{c}_2^2$	=	114
$\mathrm{c}_1\mathrm{c}_3$	=	32
$\mathrm{c}_4$	=	20

Hodge diamond

				1
			0		0
		0		2		0
	0		0		0		0
0		0		14		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 + 15	0	x₀ + 15	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 32	0	0	x₁ + 9	23
$\mathrm{HH}^{ 4 }$ = 36	0	0	0	0	36
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 31t + 82t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(473)
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( t + 1 \right) \left( 2t + 1 \right) \left( 4t^{3} + 12t^{2} + 14t - 1 \right) \left( 108t^{3} + 108t^{2} + 18t + 1 \right) \left( 51938703360t^{13} + 185112594432t^{12} + 298180206720t^{11} + 284423577408t^{10} + 258877249200t^{9} + 253110294576t^{8} + 163864206232t^{7} + 59891307652t^{6} + 11818926044t^{5} + 1252651360t^{4} - 53703714t^{3} + 15415430t^{2} + 244582t + 1241 \right)D^{6} + 2 \left( 381437837475840t^{21} + 3043718203686912t^{20} + 11263894580348928t^{19} + 25381760109972480t^{18} + 39588616406847744t^{17} + 47816209295141760t^{16} + 50281640464460544t^{15} + 48235665950417088t^{14} + 39306302122913280t^{13} + 24648744225800000t^{12} + 11207721722478096t^{11} + 3587726252463224t^{10} + 788087893240960t^{9} + 112500662903660t^{8} + 9573183617480t^{7} + 924147958522t^{6} + 180171021288t^{5} + 10311715165t^{4} - 34501455t^{3} + 30503489t^{2} + 358186t + 1241 \right)D^{5} + \left( 5160629565849600t^{21} + 36908320278675456t^{20} + 122066775852797952t^{19} + 246866194888667136t^{18} + 353553184331753472t^{17} + 414487235850882048t^{16} + 447253985924512704t^{15} + 434526831990793728t^{14} + 338404756544520480t^{13} + 194031399090810640t^{12} + 78940472844208768t^{11} + 22375359582949520t^{10} + 4285412626880928t^{9} + 497211066731152t^{8} + 30011141642932t^{7} + 4621539790872t^{6} + 780825540076t^{5} + 16038574462t^{4} + 853671728t^{3} - 58732166t^{2} - 632933t - 1241 \right)D^{4} + 2 t \left( 8862820341350400t^{20} + 57881168146022400t^{19} + 175172716774840320t^{18} + 325930461864477696t^{17} + 441320907484880640t^{16} + 518465042487599616t^{15} + 576625996418028480t^{14} + 554497176199344768t^{13} + 406900632557848128t^{12} + 214260601004292688t^{11} + 79363942223060112t^{10} + 20492457691020424t^{9} + 3570737236508880t^{8} + 376506851170092t^{7} + 34212811582528t^{6} + 8180656545458t^{5} + 867024126594t^{4} + 21737862567t^{3} + 337756835t^{2} + 2031945t + 527 \right)D^{3} + 4 t ^{2} \left( 8122382186250240t^{19} + 49325884365570048t^{18} + 139433715243270144t^{17} + 243422743107179520t^{16} + 318348866744830080t^{15} + 380788406132683392t^{14} + 432580768590229776t^{13} + 405784802124805536t^{12} + 280438950279178584t^{11} + 137191723875923580t^{10} + 47189233678137400t^{9} + 11416816178329032t^{8} + 1871025255368144t^{7} + 192751066219292t^{6} + 27114846091802t^{5} + 6042667200328t^{4} + 434426696764t^{3} + 9338995322t^{2} + 145560308t + 810957 \right)D^{2} + 8 t ^{2} \left( 3747065815203840t^{19} + 21514222986412032t^{18} + 57787422549792768t^{17} + 96167674969721856t^{16} + 123286888980011520t^{15} + 150911180637973632t^{14} + 173361826895361840t^{13} + 157833368268327936t^{12} + 103357903729420776t^{11} + 47596560862088196t^{10} + 15489905939820168t^{9} + 3598435232366440t^{8} + 570725239165408t^{7} + 62241361104720t^{6} + 11882275437826t^{5} + 2205565603466t^{4} + 115957317082t^{3} + 2436636605t^{2} + 33630055t + 161330 \right)D + 32 t ^{2} \left( 336562797772800t^{19} + 1854529651556352t^{18} + 4801338401439744t^{17} + 7716654494277120t^{16} + 9789338522124864t^{15} + 12245658890423040t^{14} + 14128299654636240t^{13} + 12503980712715120t^{12} + 7830704052705816t^{11} + 3438630919869108t^{10} + 1075822363340680t^{9} + 244350211015044t^{8} + 38209309497308t^{7} + 4590353849262t^{6} + 1030471265474t^{5} + 159426604290t^{4} + 6547018206t^{3} + 143649282t^{2} + 1666816t + 6205 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 5)

Show multiplicities

Sector dimension: 10
Spectral radius: $14.8275008$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$14.8275$	1	1
$-8.381102 - 4.124189\mathrm{i}$	1	1
$-8.381102 + 4.124189\mathrm{i}$	1	1
$-2$	1	1
$-1.237797$	1	1
$-1$	1	1
$-0.4137504 - 0.3139739\mathrm{i}$	1	1
$-0.4137504 + 0.3139739\mathrm{i}$	1	1
$0$	2	1, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda^{3} + 18 \lambda^{2} + 108 \lambda + 108\right) \left(\lambda^{3} - 14 \lambda^{2} - 12 \lambda - 4\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right) \left(\lambda + 2\right) \left(\lambda^{3} + 18 \lambda^{2} + 108 \lambda + 108\right) \left(\lambda^{3} - 14 \lambda^{2} - 12 \lambda - 4\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 12 & 48 & 12 & 30 & 72 & 40 & 100 \\ 1 & -1 & 3 & 4 & 4 & 4 & 0 & 33 & 15 & 8 \\ 1 & 1 & -1 & 6 & 20 & 6 & 10 & 9 & 3 & 24 \\ 0 & 1 & 0 & 0 & 3 & 0 & 2 & 0 & 0 & 6 \\ 0 & 1 & 2 & 0 & -1 & 0 & 0 & 10 & 6 & 6 \\ 0 & 1 & -1 & -1 & 2 & -1 & 3 & -6 & -2 & 6 \\ 0 & 0 & 1 & 1 & 1 & 1 & -1 & 6 & 2 & 0 \\ 0 & 0 & 0 & 1 & \frac{13}{2} & 1 & 5 & \frac{1}{2} & \frac{3}{2} & 10 \\ 0 & 0 & 0 & 1 & -\frac{7}{2} & 1 & -5 & \frac{1}{2} & -\frac{5}{2} & -10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta