Family 3-131-33-b

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

ℙ²P2 × ℙ²P2 × ℙ⁴P4 · 202040.wEwEw3w3 (preferred model)

Defining data

Ambient: ℙ²P2 × ℙ²P2 × ℙ⁴P4
Bundle: $(E^{ω_{3} + ω_{5}})^{\oplus 2} \oplus (E^{ω_{1} + ω_{5}})^{\oplus 2}$ 2 copies of E^(w3 + w5) + 2 copies of E^(w1 + w5)
Bundle: $(O_{2} (1) ⊠ O_{3} (1))^{\oplus 2} \oplus (O_{1} (1) ⊠ O_{3} (1))^{\oplus 2}$ 2 copies of O_2(1) box O_3(1) + 2 copies of O_1(1) box O_3(1)
ZL62 label: 202040.wEwEw3w3

Line bundles on the ambient $Y$

$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;4) \end{aligned} \end{equation}$$

Numerical invariants

Fano index	=	1
$\mathrm{h}^0(X, \omega_X^\vee)$	=	33
$\operatorname{vol}(X) = \mathrm{c}_1^4$	=	131
$\chi(X, \mathrm{T}_X)$	=	-12

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

Chern numbers of $\mathrm{T}_X$

$\mathrm{c}_1^4$	=	131
$\mathrm{c}_1^2\mathrm{c}_2$	=	122
$\mathrm{c}_2^2$	=	115
$\mathrm{c}_1\mathrm{c}_3$	=	40
$\mathrm{c}_4$	=	22

Hodge diamond

				1
			0		0
		0		3		0
	0		0		0		0
0		0		14		0		0
	0		0		0		0
		0		3		0
			0		0
				1

Polyvector parallelogram & Hochschild cohomology

$\mathrm{HH}^{ 0 }$ = 1	1
$\mathrm{HH}^{ 1 }$ = x₀	0	x₀
$\mathrm{HH}^{ 2 }$ = x₀ + x₁ + 12	0	x₀ + 12	x₁
$\mathrm{HH}^{ 3 }$ = x₁ + 24	0	0	x₁ + 12	12
$\mathrm{HH}^{ 4 }$ = 33	0	0	0	0	33
$\mathrm{HH}^{ 5 }$ = 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

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

$\displaystyle\frac{ 1 + 28t + 73t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$

$=$

Regularized quantum period

Quantum period in the literature

catalogue	matching entries
Kalashnikov (2019)	CKK(523)
Coates et al. (2020)	No match
Coates–Kasprzyk–Prince (2015)	CKP(564)

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( 3t + 1 \right) \left( 89t^{4} + 59t^{3} + 17t^{2} + 5t + 1 \right) \left( 8859t^{5} + 7412t^{4} + 2030t^{3} + 183t^{2} - 2t - 1 \right) \left( 100363208374469808t^{17} + 585450589502799720t^{16} + 1273782019575359016t^{15} + 1547550285263337312t^{14} + 1227745324668871998t^{13} + 687984220904964084t^{12} + 283453030461423849t^{11} + 87027202506891660t^{10} + 19713438797450686t^{9} + 3187950012877149t^{8} + 349108211347796t^{7} + 25276695912731t^{6} + 1693782560964t^{5} + 186742868815t^{4} + 15873045305t^{3} + 450593027t^{2} - 4220422t + 35108 \right)D^{6} + \left( 4985282736381722959006704t^{27} + 37862657097187398936180048t^{26} + 119536661623935526103303352t^{25} + 222893051666567130128768544t^{24} + 283606487005582755894984750t^{23} + 266156641904920867826144892t^{22} + 193203012555965854582841763t^{21} + 111880992138597124722048924t^{20} + 52746353596018880464478154t^{19} + 20510397097429361390217165t^{18} + 6628674184041309995610264t^{17} + 1786871403974309900994880t^{16} + 401855606189814575425410t^{15} + 75213513370055503606978t^{14} + 11677494648157441900250t^{13} + 1507643986475237908092t^{12} + 166170598231294803983t^{11} + 16873464513790708649t^{10} + 1730236259424783250t^{9} + 172992932770894570t^{8} + 14218200807907218t^{7} + 818772543758542t^{6} + 32491700649025t^{5} + 1410666352424t^{4} + 77682027275t^{3} + 1873013808t^{2} - 13363426t + 70216 \right)D^{5} + \left( 41544022803181024658389200t^{27} + 313150804819947173878324560t^{26} + 966676065937080334846824672t^{25} + 1751107371983673475463260344t^{24} + 2155529001458082263759647722t^{23} + 1949799971364743555165258892t^{22} + 1358632657714410832954183281t^{21} + 751347301621186844172142596t^{20} + 335981706971560603140490146t^{19} + 122809091473804425559366386t^{18} + 36880401496076954965557994t^{17} + 9107414042883578153663413t^{16} + 1845659233397836517974727t^{15} + 305943136001802078538209t^{14} + 41451233263196723642501t^{13} + 4648385061163991402701t^{12} + 453789175138253351834t^{11} + 42791966652968841774t^{10} + 4157850214408596187t^{9} + 374057041560384532t^{8} + 25700706102706364t^{7} + 1110556786828648t^{6} + 18071617397705t^{5} - 1010752716372t^{4} - 86260729403t^{3} - 2153016785t^{2} + 5216536t - 35108 \right)D^{4} + t \left( 174484895773360303565234640t^{26} + 1307407965976950333342482880t^{25} + 3962311683726751725549369240t^{24} + 7011306290182966765202653344t^{23} + 8405415535986392618071628802t^{22} + 7387084713036706202393228484t^{21} + 4988371925589074795986446573t^{20} + 2664673263291436494993606276t^{19} + 1145691096308363301246345126t^{18} + 400083748334517704990858361t^{17} + 113816434820323838982407964t^{16} + 26352391359073522332944756t^{15} + 4954238586395637458884328t^{14} + 757104866928749386966264t^{13} + 95565197242513556183826t^{12} + 10512778931587892755532t^{11} + 1117202447502895731741t^{10} + 122894193565093161891t^{9} + 12912905099965544230t^{8} + 1106304864425392572t^{7} + 67133024293842496t^{6} + 2476894961222972t^{5} + 40516009397997t^{4} - 9841910182t^{3} + 6387552381t^{2} + 217914678t + 64588 \right)D^{3} + 2 t ^{2} \left( 192764265806759954414925888t^{25} + 1437760176544240682749789428t^{24} + 4294497367181793908979238788t^{23} + 7460396988887448138885261444t^{22} + 8761446568004509993240312452t^{21} + 7530698779997823891404163810t^{20} + 4965112709972909928558598671t^{19} + 2583604078392341858151750912t^{18} + 1078421290031881091061182814t^{17} + 363820525908837587599177695t^{16} + 99337619445322903880624840t^{15} + 21903305064904118817847938t^{14} + 3893007619767714558701028t^{13} + 561701441640439599050965t^{12} + 68114231131059763040333t^{11} + 7567055864542803849317t^{10} + 853396725759094657750t^{9} + 96509498692573620394t^{8} + 9557389107329060376t^{7} + 722922363138528500t^{6} + 37248762590855048t^{5} + 1122529486994861t^{4} + 13927638768106t^{3} + 34342592175t^{2} + 6896679192t + 69855619 \right)D^{2} + 4 t ^{2} \left( 104690937464016182139140784t^{25} + 778220018237557179707153148t^{24} + 2299271255275455362953266372t^{23} + 3939503801565831470724597480t^{22} + 4555843933340395044726805566t^{21} + 3851613088135776039962485638t^{20} + 2494763879849723520972694800t^{19} + 1273166851466391708564387486t^{18} + 519852474081921805294188789t^{17} + 170905583786133569143049592t^{16} + 45242791149644129977880424t^{15} + 9615370145995067473013121t^{14} + 1639680081322004857751107t^{13} + 227561772121801493080639t^{12} + 27124926778842937057979t^{11} + 3092575981766772655637t^{10} + 363497235772479574299t^{9} + 40779756512154708981t^{8} + 3766773181723783231t^{7} + 254988357036756539t^{6} + 11400945669630735t^{5} + 287012918353808t^{4} + 2852899622504t^{3} + 25198443253t^{2} + 2162897088t + 11655856 \right)D + 24 t ^{2} \left( 7121832480545318512866720t^{25} + 52819150679938524949984440t^{24} + 154890055252765766090161512t^{23} + 262871613869174859964749660t^{22} + 300786370924143821457406032t^{21} + 251401820828183275028074596t^{20} + 160849783704771759786047892t^{19} + 80985488548483605552918606t^{18} + 32560951450613072174866035t^{17} + 10510616329574139418803060t^{16} + 2721562858268267844914439t^{15} + 563330318063462473797939t^{14} + 93287443688688604509424t^{13} + 12622524072285884539207t^{12} + 1497079886567030727380t^{11} + 175161902176320227591t^{10} + 21070517918139326778t^{9} + 2318149614505377837t^{8} + 201511407497212297t^{7} + 12467927301570872t^{6} + 495974773518342t^{5} + 10732103171813t^{4} + 96039570455t^{3} + 1931462455t^{2} + 84973956t + 280864 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ²P2 × ℙ²P2 × ℙ⁴P4

Show multiplicities

Sector dimension: 14
Spectral radius: $17.1478794$
Diagonalizable at 1: Yes
Simple spectrum at 1: No

Eigenvalue approximations and Jordan block sizes

eigenvalue	multiplicity	Jordan blocks
$17.14788$	1	1
$-6.594942 - 3.785884\mathrm{i}$	1	1
$-6.594942 + 3.785884\mathrm{i}$	1	1
$0.206542 - 3.383479\mathrm{i}$	2	1, 1
$0.206542 + 3.383479\mathrm{i}$	2	1, 1
$-3$	1	1
$-2.978998 - 0.2442346\mathrm{i}$	1	1
$-2.978998 + 0.2442346\mathrm{i}$	1	1
$-2.706542 - 0.6481502\mathrm{i}$	2	1, 1
$-2.706542 + 0.6481502\mathrm{i}$	2	1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 5 \lambda^{3} + 17 \lambda^{2} + 59 \lambda + 89\right)^{2} \left(\lambda^{5} + 2 \lambda^{4} - 183 \lambda^{3} - 2030 \lambda^{2} - 7412 \lambda - 8859\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 5 \lambda^{3} + 17 \lambda^{2} + 59 \lambda + 89\right) \left(\lambda^{5} + 2 \lambda^{4} - 183 \lambda^{3} - 2030 \lambda^{2} - 7412 \lambda - 8859\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 12 & 27 & 39 & 15 & 39 & 15 & 39 & 160 & 60 & 60 & 420 \\ 1 & -1 & 0 & 2 & 6 & 10 & 2 & 12 & 4 & 14 & 30 & 12 & 9 & 80 \\ 1 & 0 & -1 & 2 & 6 & 12 & 4 & 10 & 2 & 14 & 30 & 9 & 12 & 80 \\ 1 & 2 & 2 & -1 & 0 & 4 & 2 & 4 & 2 & 4 & 36 & 15 & 15 & 80 \\ 0 & 1 & 1 & 0 & -2 & 2 & 0 & 2 & 0 & 4 & 8 & 2 & 2 & 12 \\ 0 & 1 & 0 & 1 & 2 & -2 & 0 & 0 & 0 & 0 & 4 & 2 & 0 & 12 \\ 0 & 1 & 0 & 0 & 0 & 2 & -1 & 0 & 0 & 1 & 12 & 4 & 6 & 24 \\ 0 & 0 & 1 & 1 & 2 & 0 & 0 & -2 & 0 & 0 & 4 & 0 & 2 & 12 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & -1 & 1 & 12 & 6 & 4 & 24 \\ 0 & 0 & 0 & 1 & 0 & 2 & 1 & 2 & 1 & -1 & 0 & 0 & 0 & 12 \\ 0 & 0 & 0 & 0 & 1 & \frac{15}{4} & \frac{3}{4} & \frac{15}{4} & \frac{3}{4} & \frac{23}{4} & 5 & \frac{7}{2} & \frac{7}{2} & 12 \\ 0 & 0 & 0 & 0 & 1 & -\frac{9}{2} & -\frac{3}{2} & \frac{1}{2} & \frac{3}{2} & -\frac{9}{2} & -4 & -4 & -3 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{1}{2} & \frac{3}{2} & -\frac{9}{2} & -\frac{3}{2} & -\frac{9}{2} & -4 & -3 & -4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{4} & \frac{3}{4} & 0\end{pmatrix}\notag\end{equation}$$

Fano fourfolds of type A beta