Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-114-30-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 7)
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_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
4 copies of E^(w4) + E^(w2 + w3)
Bundle
4 copies of O_2(1) + Q_1 box U_2^v
ZL62 label
20601.3333w7
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 30
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 114
$\chi(X, \mathrm{T}_X)$ = -20

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 114
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 117
$\mathrm{c}_1\mathrm{c}_3$ = 26
$\mathrm{c}_4$ = 23
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 17 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 + 20 0 x0 + 20 x2
$\mathrm{HH}^{ 3 }$ = x1 + 12 0 0 x1 12
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 14 0 0 x1 - x2 - 16 0 30
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 25t + 62t^{2} + 25t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(527)
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( 2t + 1 \right) \left( 3t + 1 \right) \left( 530696t^{8} + 739624t^{7} + 409826t^{6} + 111999t^{5} + 15031t^{4} + 983t^{3} + 97t^{2} + 9t - 1 \right) \left( 337379501242219520t^{17} + 1332041344763888640t^{16} + 2266494840491290368t^{15} + 2275128843858160128t^{14} + 1551033405110687744t^{13} + 788276006704054336t^{12} + 322517429865660400t^{11} + 111935840806371040t^{10} + 32983941362055150t^{9} + 7859997931781476t^{8} + 1421169263496925t^{7} + 184518501489915t^{6} + 16490327257600t^{5} + 987353216418t^{4} + 39106388456t^{3} + 908762232t^{2} + 719789t + 15447 \right)D^{6} + \left( 22559789925696357228625920t^{27} + 133983361259494364080906240t^{26} + 365345712246809643212324864t^{25} + 614463797667538825414256640t^{24} + 721391777155665442628218880t^{23} + 634500580227676678174600192t^{22} + 438839314697871335282184704t^{21} + 247802592373626465940924288t^{20} + 117764580318723335349824224t^{19} + 48146393822945276902703760t^{18} + 17108717344194877744141352t^{17} + 5270270705650992946998256t^{16} + 1389911049240483016229774t^{15} + 308539353232104628741796t^{14} + 56671452033594603197685t^{13} + 8483073710154503924704t^{12} + 1022354461877352556240t^{11} + 98580490002089770346t^{10} + 7682652265805894446t^{9} + 510926611241704280t^{8} + 32752383175544172t^{7} + 2190046965907306t^{6} + 134799723606824t^{5} + 6276359412838t^{4} + 196256502140t^{3} + 3650532836t^{2} + 2282943t + 30894 \right)D^{5} + \left( 187998249380802976905216000t^{27} + 1081006141267369516542156800t^{26} + 2838110422787057083288903680t^{25} + 4583114772646458289905018880t^{24} + 5163524706608097819983564800t^{23} + 4366377589608007600506549248t^{22} + 2916771431041311512436312576t^{21} + 1602341252603410915442105728t^{20} + 747026512987546664575271840t^{19} + 301442949336653514294605296t^{18} + 105777358929828468054528312t^{17} + 31988031758158193863828016t^{16} + 8198399533454269537180378t^{15} + 1748497248618916377356180t^{14} + 305157490345377399225955t^{13} + 42941855635118719600427t^{12} + 4804044872243753203600t^{11} + 420232015121063270300t^{10} + 27923078747747902132t^{9} + 1300975289444380956t^{8} + 28811115361839768t^{7} - 1297101776941472t^{6} - 157978381657976t^{5} - 7245830943648t^{4} - 188519724466t^{3} - 2979916912t^{2} - 1045815t - 15447 \right)D^{4} + t \left( 789592647399372503001907200t^{26} + 4423003624869176361841254400t^{25} + 11255296655320887107025469440t^{24} + 17577894786927717846761461760t^{23} + 19153871260968930447992221696t^{22} + 15702782238983691053376556032t^{21} + 10220074389041458627298387456t^{20} + 5508666867352428570430438016t^{19} + 2537172020687470503399671712t^{18} + 1014737232123722283340496176t^{17} + 352000309773135752632818168t^{16} + 104437251252385319581469712t^{15} + 26009114234094258855663882t^{14} + 5340700937972643032565220t^{13} + 890825783627082761546155t^{12} + 119226942358250634526364t^{11} + 12668270781032752773880t^{10} + 1057568406959923941318t^{9} + 68460203557078029850t^{8} + 3354159771109421576t^{7} + 117536936203163828t^{6} + 2547818952278176t^{5} + 19732125344232t^{4} - 354486015910t^{3} - 9659797872t^{2} + 40288252t + 38753 \right)D^{3} + 2 t ^{2} \left( 872311877126925812840202240t^{25} + 4787285266995732121865789440t^{24} + 11882751331128916198856585216t^{23} + 18070033167064037853352110080t^{22} + 19179201437544053929106988032t^{21} + 15353288583022086547476977664t^{20} + 9802865811616216753226943232t^{19} + 5214946055344352710584576384t^{18} + 2382547542693256066659489856t^{17} + 946147393092119097341481664t^{16} + 324541363342524293390431280t^{15} + 94502592602575159976088696t^{14} + 22907371031885246753334248t^{13} + 4544862293994316412468952t^{12} + 728290943606223868636275t^{11} + 93272146761694879921210t^{10} + 9460465707061141017998t^{9} + 752531878989717627582t^{8} + 46198503721199998096t^{7} + 2110325090118629472t^{6} + 65461108423446915t^{5} + 1069708410060542t^{4} + 567599762310t^{3} - 118525297468t^{2} + 696291984t + 17181852 \right)D^{2} + 8 t ^{2} \left( 236877794219811750900572160t^{25} + 1280278634098455791879045120t^{24} + 3118408766488846813580183552t^{23} + 4647126703650036199775805440t^{22} + 4835409697290111491695299072t^{21} + 3802933623384998437612072448t^{20} + 2394917155387551145594276224t^{19} + 1262675722655246348400284672t^{18} + 573678335801910943220118352t^{17} + 226481919848023133923074216t^{16} + 76900151130870186245655388t^{15} + 22023117694803606177011516t^{14} + 5215832397533511790545171t^{13} + 1005419992181336133844482t^{12} + 155873000940978735222563t^{11} + 19259296300946561704034t^{10} + 1881697547325476891247t^{9} + 143992474514216968528t^{8} + 8464669935453503158t^{7} + 363703777036998020t^{6} + 9979853153314123t^{5} + 104922829402382t^{4} - 1827461706937t^{3} - 30871897370t^{2} + 265081943t + 1977216 \right)D + 96 t ^{2} \left( 8057067830605841867366400t^{25} + 43093807249585016082636800t^{24} + 103597712241912520213514240t^{23} + 152216651036543792427745280t^{22} + 156202245191246946474948096t^{21} + 121351582295711755633111040t^{20} + 75710599813270758134897280t^{19} + 39682089924132868277649536t^{18} + 17961707387546419776956464t^{17} + 7059265900773400757822712t^{16} + 2377599348107946535086916t^{15} + 672081940315273155008828t^{14} + 156337345391061299746089t^{13} + 29476213694069361300918t^{12} + 4455523510751279167226t^{11} + 535610172721141958300t^{10} + 50853982274417266494t^{9} + 3777687284358255064t^{8} + 214787316487424842t^{7} + 8804654541081104t^{6} + 219489201472666t^{5} + 1367679829700t^{4} - 67909078054t^{3} - 962535080t^{2} + 3189941t + 30894 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 7)

|λ| = 19.51011 λ ≈ 19.51011; multiplicity 1; Jordan blocks 1 λ ≈ 2.28468 - 8.82792i; multiplicity 1; Jordan blocks 1 λ ≈ 2.28468 + 8.82792i; multiplicity 1; Jordan blocks 1 λ ≈ -3.039548 - 1.623978i; multiplicity 1; Jordan blocks 1 λ ≈ -3.039548 + 1.623978i; multiplicity 1; Jordan blocks 1 λ ≈ -3.15555 - 0.5337856i; multiplicity 1; Jordan blocks 1 λ ≈ -3.15555 + 0.5337856i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.689273; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$19.5101099$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.51011$11
$2.28468 - 8.82792\mathrm{i}$11
$2.28468 + 8.82792\mathrm{i}$11
$-3.039548 - 1.623978\mathrm{i}$11
$-3.039548 + 1.623978\mathrm{i}$11
$-3.15555 - 0.5337856\mathrm{i}$11
$-3.15555 + 0.5337856\mathrm{i}$11
$-3$11
$-2.689273$11
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{8} - 9 \lambda^{7} - 97 \lambda^{6} - 983 \lambda^{5} - 15031 \lambda^{4} - 111999 \lambda^{3} - 409826 \lambda^{2} - 739624 \lambda - 530696\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{8} - 9 \lambda^{7} - 97 \lambda^{6} - 983 \lambda^{5} - 15031 \lambda^{4} - 111999 \lambda^{3} - 409826 \lambda^{2} - 739624 \lambda - 530696\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 22 & 66 & 33 & 63 & 126 & 344 & 624 & 1650 \\ 1 & 0 & 4 & 12 & -6 & 18 & 28 & 0 & -9 & -40 \\ 1 & 0 & 2 & 8 & 6 & 10 & 20 & 63 & 117 & 360 \\ 0 & 1 & 1 & 2 & -1 & 3 & 6 & 2 & 0 & -30 \\ 0 & 1 & 0 & 4 & 1 & 1 & 4 & 20 & 36 & 75 \\ 0 & 0 & 1 & 0 & 1 & -3 & 0 & 0 & 0 & 15 \\ 0 & 0 & 1 & 0 & 0 & 2 & 0 & 6 & 12 & 45 \\ 0 & 0 & 0 & -4 & -5 & 7 & 8 & -18 & -30 & -80 \\ 0 & 0 & 0 & \frac{13}{3} & \frac{10}{3} & -\frac{7}{3} & -\frac{4}{3} & 12 & 20 & \frac{170}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{5} & \frac{12}{5} & 0\end{pmatrix}\notag\end{equation}$$