Family 2-165-40-a

← 2-176-42-d · 2-163-40-a →

No geometric description is recorded for this family (yet).

Varieties in this family can be blown up to

3-135-34-a

Varieties in this family can be blown down to

1-405-85-a

ℙ²P2 × Gr(2, 5) · 2041.33m (preferred model)

Defining data

Ambient: ℙ²P2 × Gr(2, 5)
Bundle: $(E^{ω_{4}})^{\oplus 2} \oplus E^{ω_{2} + ω_{4}}$ 2 copies of E^(w4) + E^(w2 + w4)
Bundle: $O_{2} (1)^{\oplus 2} \oplus Q_{1} ⊠ O_{2} (1)$ 2 copies of O_2(1) + Q_1 box O_2(1)
ZL62 label: 2041.33m

Line bundles on the ambient $Y$

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

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	40
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	165
$\chi(X, \mathrm{T}_X)$	=	-7

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	165
$\mathrm{c}_1^2\mathrm{c}_2$	=	138
$\mathrm{c}_2^2$	=	97
$\mathrm{c}_1\mathrm{c}_3$	=	48
$\mathrm{c}_4$	=	6

Hodge diamond

				1
			0		0
		0		2		0
	0		1		1		0
0		0		4		0		0
	0		1		1		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 + 7	0	x₀ + 7	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 42	0	0	x₁ + 19	23
$\mathrm{HH}^{ 4 }$ = 40	0	0	0	0	40
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 35t + 93t^{2} + 35t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(474)
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( 13t^{2} + 6t + 1 \right) ^{2} \left( 26479t^{6} + 17475t^{5} + 2835t^{4} - 328t^{3} + 15t - 1 \right) \left( 16731519993264791t^{15} + 37951385732429382t^{14} + 43074620075543897t^{13} + 34451970512243077t^{12} + 20820459203758737t^{11} + 9196625593719409t^{10} + 2843672535171248t^{9} + 597763438470273t^{8} + 84153752468810t^{7} + 8035795084190t^{6} + 565739189457t^{5} + 31727041265t^{4} + 1069431230t^{3} + 12409038t^{2} - 1100332t + 528 \right)D^{6} + 3 \left( 13t^{2} + 6t + 1 \right) \left( 40316086529050914480899t^{23} + 129745725824367137402203t^{22} + 205107693596504640734415t^{21} + 217792407549447303703984t^{20} + 174157006737975175037544t^{19} + 108328971234381871884107t^{18} + 52262734450948382849824t^{17} + 19327484523460592025337t^{16} + 5417651764846260517057t^{15} + 1139850013997911677486t^{14} + 178436577219624488531t^{13} + 20635151363096026816t^{12} + 1755185887268753699t^{11} + 112407807534464167t^{10} + 7488638268285783t^{9} + 1046675978352559t^{8} + 164389660609443t^{7} + 16508674103201t^{6} + 1088693594335t^{5} + 55908374632t^{4} + 1496272474t^{3} + 19961484t^{2} - 1100332t + 352 \right)D^{5} + \left( 13102728121941547206292175t^{25} + 46416660229130148867602745t^{24} + 81162071190241294002421516t^{23} + 95322116987046526004917610t^{22} + 84422087385600994418807432t^{21} + 58726895852619584069896297t^{20} + 32335319497914432287660987t^{19} + 14056569706178808079809767t^{18} + 4807770810301109128870102t^{17} + 1290521778822668792853185t^{16} + 271310568395695270147136t^{15} + 44591110252525827646216t^{14} + 5717406753867701994742t^{13} + 568070566785228787067t^{12} + 42181346524753953215t^{11} + 1902463447515144530t^{10} - 41494611537888851t^{9} - 19063079802263831t^{8} - 2145937821978686t^{7} - 148425424050733t^{6} - 7149508390240t^{5} - 261634436727t^{4} - 5314766024t^{3} - 52632198t^{2} + 2103096t - 528 \right)D^{4} + t \left( 55031458112154498266427135t^{24} + 189016197939962158975112865t^{23} + 321953787630976192936117120t^{22} + 371355933451641453703869129t^{21} + 324613490406376393499870847t^{20} + 222541783974185095839626047t^{19} + 120028496358758472497826966t^{18} + 50746398239531478773306196t^{17} + 16780537334086629751841715t^{16} + 4339805350982977287710919t^{15} + 879260474634880453439990t^{14} + 140076877287741529391064t^{13} + 17668308776118261560960t^{12} + 1780339236397290400097t^{11} + 143569713873645376740t^{10} + 9004399254578347178t^{9} + 397509257365683784t^{8} + 8506747311198230t^{7} - 284884564194842t^{6} - 33652420169615t^{5} - 1345541695025t^{4} - 44894854214t^{3} - 1215281326t^{2} + 9087096t + 944 \right)D^{3} + 2 t ^{2} \left( 60796658485808779037195692t^{23} + 203802442169514181521480342t^{22} + 340159742390085194626747906t^{21} + 387107095152301593203574106t^{20} + 335016028267866564799639905t^{19} + 226875374261466166752117211t^{18} + 120180747245009386615516090t^{17} + 49591334045519334010108851t^{16} + 15925231743143783517859378t^{15} + 3988747863740186258286082t^{14} + 783009088701591888434715t^{13} + 121461260015044803881158t^{12} + 15073530001889760414332t^{11} + 1516294155359990835867t^{10} + 123610936987488651893t^{9} + 7859002885435056799t^{8} + 347614490995906239t^{7} + 7011525184918648t^{6} - 276456649719676t^{5} - 27209630668950t^{4} - 887721113842t^{3} - 15411000024t^{2} - 274763292t + 1890570 \right)D^{2} + 4 t ^{2} \left( 33018874867292698959856281t^{23} + 108689670450716924489416509t^{22} + 178695108719233094551342286t^{21} + 201391336049037679336973871t^{20} + 173006963241768401712114870t^{19} + 116028836012144998617418994t^{18} + 60572211759616821485351282t^{17} + 24507349521808824293000481t^{16} + 7686197351463784227660040t^{15} + 1876342810693583736811502t^{14} + 359254051181715884167838t^{13} + 54595891270600007069164t^{12} + 6693260782804051741750t^{11} + 671393054098462042121t^{10} + 54805114662115087751t^{9} + 3469362757764082408t^{8} + 150207941497119418t^{7} + 2740307597609532t^{6} - 134634444031882t^{5} - 11337228738668t^{4} - 333152665116t^{3} - 3474332328t^{2} - 16981504t + 158400 \right)D + 16 t ^{2} \left( 3369272945642112138760845t^{23} + 10953171961056818216248005t^{22} + 17822846081103899738491914t^{21} + 19955429688106426194045099t^{20} + 17055937966076929824829924t^{19} + 11359100635818285381155130t^{18} + 5867122681075224777724199t^{17} + 2339845589365821791797780t^{16} + 721219087834757677175201t^{15} + 172775371813632767123096t^{14} + 32483329700174321209926t^{13} + 4864321618597929353687t^{12} + 591206469646245238096t^{11} + 59127863544561437154t^{10} + 4814636191129586974t^{9} + 302150019992142442t^{8} + 12836329829282868t^{7} + 223435615579609t^{6} - 11341086356689t^{5} - 879380683144t^{4} - 23811009838t^{3} - 185781654t^{2} + 1294096t + 5280 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × Gr(2, 5)

Show multiplicities

Sector dimension: 10
Spectral radius: $14.7802912$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$14.78029$	1	1
$4.745109 - 5.793815\mathrm{i}$	1	1
$4.745109 + 5.793815\mathrm{i}$	1	1
$-4.435437$	1	1
$-3 - 2\mathrm{i}$	2	1, 1
$-3 + 2\mathrm{i}$	2	1, 1
$-2.417536 - 1.165035\mathrm{i}$	1	1
$-2.417536 + 1.165035\mathrm{i}$	1	1

Characteristic polynomial

$$\begin{equation}\left(\lambda^{2} + 6 \lambda + 13\right)^{2} \left(\lambda^{6} - 15 \lambda^{5} + 328 \lambda^{3} - 2835 \lambda^{2} - 17475 \lambda - 26479\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{2} + 6 \lambda + 13\right) \left(\lambda^{6} - 15 \lambda^{5} + 328 \lambda^{3} - 2835 \lambda^{2} - 17475 \lambda - 26479\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 20 & 9 & 9 & 33 & 51 & 40 & 80 & 150 \\ 2 & 0 & 3 & 20 & -2 & 12 & 18 & 30 & 0 & 8 \\ 1 & 0 & 2 & 0 & 2 & 8 & 12 & 6 & 15 & 32 \\ 0 & 1 & 2 & 2 & 0 & 2 & 3 & 8 & -2 & 0 \\ 0 & 2 & 0 & 3 & 0 & 1 & 2 & 12 & 20 & 30 \\ 0 & 0 & 1 & 0 & 0 & -3 & 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 2 & 6 \\ 0 & 0 & 0 & \frac{5}{2} & 0 & 3 & \frac{9}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta