Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-90-25-b

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3 + w4) + E^(w1 + w3)
Bundle
U_2^v(1) + O_1(1) box U_2^v
ZL62 label
2041.ph
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 25
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 90
$\chi(X, \mathrm{T}_X)$ = -21

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 90
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 127
$\mathrm{c}_1\mathrm{c}_3$ = 22
$\mathrm{c}_4$ = 25
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 19 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 21 0 x0 + 21 x2
$\mathrm{HH}^{ 3 }$ = x1 + x3 0 0 x1 x3
$\mathrm{HH}^{ 4 }$ = x1 - x2 + x3 + 3 0 0 x1 - x2 - 17 x3 - 5 25
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 20t + 48t^{2} + 20t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(617)
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( 3t + 1 \right) \left( 4t + 1 \right) ^{2} \left( 148652t^{7} + 190047t^{6} + 90314t^{5} + 20016t^{4} + 2381t^{3} + 218t^{2} + 10t - 1 \right) \left( 999957461291714944t^{17} + 4412610196608056832t^{16} + 8705341686583318032t^{15} + 10253440930846911808t^{14} + 8089491793065245400t^{13} + 4532175950551283280t^{12} + 1861439745238973949t^{11} + 569890839738722400t^{10} + 130968040872912696t^{9} + 22606099449864934t^{8} + 2922473661230448t^{7} + 282274174641915t^{6} + 20396021622517t^{5} + 1100267057901t^{4} + 42157485543t^{3} + 914735274t^{2} + 2906901t + 8370 \right)D^{6} + \left( 4t + 1 \right) \left( 37458710487055874483582976t^{26} + 227842065550942637393142784t^{25} + 643517469973800084827885184t^{24} + 1126107153970109513826048000t^{23} + 1372577989333436103694162800t^{22} + 1240731228535729875895938800t^{21} + 864091843160751528516625192t^{20} + 475247328816625656353258256t^{19} + 209859994162531848382612183t^{18} + 75231657930015010156339043t^{17} + 22054291285673331834880178t^{16} + 5311146669063672995241209t^{15} + 1053549156387237199689757t^{14} + 172430679385513430604312t^{13} + 23325175312836540696122t^{12} + 2617152474546568532380t^{11} + 245516780682616680693t^{10} + 19547412305323477276t^{9} + 1348453343389193612t^{8} + 81794735207077762t^{7} + 4324604105210192t^{6} + 192962008464340t^{5} + 7034829115404t^{4} + 200859031683t^{3} + 3635949543t^{2} + 8637003t + 16740 \right)D^{5} + \left( 1248623682901862482786099200t^{27} + 7714988104660983282544547840t^{26} + 22154517319552474141551307776t^{25} + 39470656841180386629450922368t^{24} + 49065890664879715084048627840t^{23} + 45327329718951259819155307600t^{22} + 32336904292701895055472397296t^{21} + 18266379522178359849817408728t^{20} + 8308416119570481234266681360t^{19} + 3077738751459767351135641677t^{18} + 935567728295395326628521181t^{17} + 234493889841284965226877850t^{16} + 48596544005711772613180725t^{15} + 8338124142604312899177311t^{14} + 1184766416891038105142791t^{13} + 139308567176333728093155t^{12} + 13520565938959986304911t^{11} + 1075337719428137037215t^{10} + 68766597061343573367t^{9} + 3372299123094199661t^{8} + 111562228530300358t^{7} + 1288209493812357t^{6} - 92819441422991t^{5} - 6104940667881t^{4} - 182157861159t^{3} - 2899607241t^{2} - 5883273t - 8370 \right)D^{4} + t \left( 5244219468187822427701616640t^{26} + 31769669035959647957539870720t^{25} + 89182997050356784532371176960t^{24} + 154942571598698215277610224512t^{23} + 187409603313803031376701837632t^{22} + 168096071317564053229587261264t^{21} + 116180863012972402927712947248t^{20} + 63436401473435748729116311544t^{19} + 27823596844106944934760415300t^{18} + 9913995821144713389970732541t^{17} + 2891386434805628256665147125t^{16} + 693584263237564869905304298t^{15} + 137263565244062116511790127t^{14} + 22455233288294513215441723t^{13} + 3040854952068127136690352t^{12} + 341255866874864771545122t^{11} + 31745685098487601732598t^{10} + 2440648305635746933081t^{9} + 153311793577277046620t^{8} + 7651236934202373270t^{7} + 286882971033498112t^{6} + 7277396509799484t^{5} + 99341912450942t^{4} + 215687975286t^{3} - 5231733135t^{2} + 27207057t + 27621 \right)D^{3} + 2 t ^{2} \left( 5793613888664641920127500288t^{25} + 34562646848574822864169603072t^{24} + 95305859068943386040288055040t^{23} + 162311002188173941091794770688t^{22} + 192084146179126791306367201888t^{21} + 168263134754091084108907613456t^{20} + 113369920457608048391274472432t^{19} + 60229272897722758677294446432t^{18} + 25652803127152588469209995298t^{17} + 8858417819329502847865127363t^{16} + 2498914075499284718077462293t^{15} + 578769045319483024430861116t^{14} + 110430977645454843166439097t^{13} + 17401202017762669187460532t^{12} + 2269054935865002392334023t^{11} + 245192111705334406487650t^{10} + 21941526329392247529785t^{9} + 1615229250323446567816t^{8} + 95942987898741938126t^{7} + 4412468001549866324t^{6} + 145470969408934482t^{5} + 2968781997730676t^{4} + 26195187052353t^{3} + 12947239791t^{2} + 3002351175t + 16477740 \right)D^{2} + 4 t ^{2} \left( 3146531680912693456620969984t^{25} + 18558055168699041864974989312t^{24} + 50492460650302286237258443520t^{23} + 84704217503937689101984479872t^{22} + 98589140994051854271568214368t^{21} + 84811233950666984770923634912t^{20} + 56030555089317729760540499552t^{19} + 29141478785641431483624895912t^{18} + 12131395586487442479093470778t^{17} + 4087864454334179437154370660t^{16} + 1123513817260467667237637726t^{15} + 253176181126456444771962447t^{14} + 46951756119694228151956433t^{13} + 7187051435476621456724278t^{12} + 910352441233975135747837t^{11} + 95557823223569338612535t^{10} + 8295727630528282627600t^{9} + 589347220145103740481t^{8} + 33341697728645502531t^{7} + 1421272058951318938t^{6} + 41034865102991332t^{5} + 625572524622186t^{4} + 582745491858t^{3} - 52108923270t^{2} + 1017684864t + 4084560 \right)D + 8 t ^{2} \left( 642149322635243562575708160t^{25} + 3757985238342268199292334080t^{24} + 10130792338764877802139683328t^{23} + 16817961179153672785899867648t^{22} + 19348233869543872343970452544t^{21} + 16432566923484331141009909120t^{20} + 10705257757230081770481832384t^{19} + 5483544234583770482431195344t^{18} + 2245305278225954778974446828t^{17} + 743203120674781692410000188t^{16} + 200396279530527810335889588t^{15} + 44254183470238272643820217t^{14} + 8036134909785373504784667t^{13} + 1204005972855112766128116t^{12} + 149263014543296760532783t^{11} + 15333173546779654194034t^{10} + 1300972201258372231647t^{9} + 89909822218052616229t^{8} + 4893559652079407188t^{7} + 196103556305709555t^{6} + 5043604522298688t^{5} + 54685452804495t^{4} - 572582399265t^{3} - 12371521644t^{2} + 72023850t + 334800 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 24.49926 λ ≈ 24.49926; multiplicity 1; Jordan blocks 1 λ ≈ -0.532785 - 7.573487i; multiplicity 1; Jordan blocks 1 λ ≈ -0.532785 + 7.573487i; multiplicity 1; Jordan blocks 1 λ ≈ -5.180402; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ -3.268916; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.492184 - 0.07125734i; multiplicity 1; Jordan blocks 1 λ ≈ -2.492184 + 0.07125734i; multiplicity 1; Jordan blocks 1 2
Sector dimension
10
Spectral radius
$24.4992551$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$24.49926$11
$-0.532785 - 7.573487\mathrm{i}$11
$-0.532785 + 7.573487\mathrm{i}$11
$-5.180402$11
$-4$22
$-3.268916$11
$-3$11
$-2.492184 - 0.07125734\mathrm{i}$11
$-2.492184 + 0.07125734\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{2} \left(\lambda^{7} - 10 \lambda^{6} - 218 \lambda^{5} - 2381 \lambda^{4} - 20016 \lambda^{3} - 90314 \lambda^{2} - 190047 \lambda - 148652\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{2} \left(\lambda^{7} - 10 \lambda^{6} - 218 \lambda^{5} - 2381 \lambda^{4} - 20016 \lambda^{3} - 90314 \lambda^{2} - 190047 \lambda - 148652\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 30 & 168 & 54 & 144 & 162 & 880 & 1056 & 3500 \\ 1 & -1 & 5 & 20 & 0 & 26 & 32 & 90 & 93 & 400 \\ 1 & 1 & 2 & 24 & 10 & 22 & 22 & 147 & 186 & 640 \\ 0 & 1 & 1 & 1 & 0 & 5 & 7 & 12 & 8 & 45 \\ 0 & 1 & 0 & 5 & -1 & 0 & 0 & 26 & 32 & 75 \\ 0 & 0 & 1 & 1 & 1 & 0 & 2 & 10 & 14 & 60 \\ 0 & 0 & 1 & 1 & 0 & 0 & -3 & 10 & 14 & 45 \\ 0 & 0 & 0 & \frac{1}{11} & -\frac{10}{11} & \frac{36}{11} & \frac{35}{11} & -\frac{36}{11} & -\frac{13}{11} & \frac{50}{11} \\ 0 & 0 & 0 & \frac{26}{11} & \frac{15}{11} & -\frac{10}{11} & -\frac{3}{11} & \frac{65}{11} & \frac{47}{11} & \frac{200}{11} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{5} & \frac{9}{5} & 0\end{pmatrix}\notag\end{equation}$$