Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-68-21-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ6P6
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_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w1 + w4) + E^(w1 + 2w4) + E^(w3 + w4)
Bundle
O_1(1) box O_2(1) + O_1(1) box O_2(2) + Q_1 box O_2(1)
ZL62 label
30600.wFw2x321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 68
$\chi(X, \mathrm{T}_X)$ = -51

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 68
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 161
$\mathrm{c}_1\mathrm{c}_3$ = -18
$\mathrm{c}_4$ = 93
Hodge diamond
1
00
0 2 0
0 0 0 0
0 8 71 8 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 + x1 + 51 0 x0 + 51 x1
$\mathrm{HH}^{ 3 }$ = x1 + 8 0 0 x1 + 1 7
$\mathrm{HH}^{ 4 }$ = 49 0 0 28 0 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 34t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(648)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(656)

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) ^{4} \left( 28757t^{5} + 32411t^{4} + 10077t^{3} + 510t^{2} + 3t - 1 \right) \left( 464798974894t^{10} + 1312202487675t^{9} + 1615891060282t^{8} + 1118944617442t^{7} + 487597329139t^{6} + 155892623137t^{5} + 44268654337t^{4} + 10022667699t^{3} + 1126198690t^{2} + 7842591t + 22100 \right)D^{6} + \left( 2t + 1 \right) ^{3} \left( 561381413083123836t^{16} + 2279466483803133720t^{15} + 4190260579717560403t^{14} + 4586523479391237873t^{13} + 3334097954806293031t^{12} + 1728532574617212458t^{11} + 688741735018065784t^{10} + 227568813055927788t^{9} + 63191348354719554t^{8} + 13412082934307903t^{7} + 1851013056827029t^{6} + 133411958186678t^{5} + 3336000383084t^{4} + 53154461769t^{3} + 4441085713t^{2} + 23174173t + 44200 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 9356356884718730600t^{17} + 39751850555084615620t^{16} + 77049596849705288850t^{15} + 89736956991962995447t^{14} + 70139075017688067619t^{13} + 39479769012240139755t^{12} + 17146508637676255511t^{11} + 6156496752134783561t^{10} + 1871617489746951610t^{9} + 453016958362525563t^{8} + 77489196472707729t^{7} + 8078902157026410t^{6} + 411896642955077t^{5} + 6513706403929t^{4} + 25253248465t^{3} - 4005933583t^{2} - 19318097t - 22100 \right)D^{4} + t \left( 2t + 1 \right) \left( 78593397831637337040t^{17} + 353957166304171382640t^{16} + 732027941481432185080t^{15} + 916911455737639937624t^{14} + 777405685500782633409t^{13} + 477810862783039994355t^{12} + 226536125951042802141t^{11} + 88144680428074293402t^{10} + 29092955401057900776t^{9} + 7887085915016516778t^{8} + 1609512219335484920t^{7} + 221315124422959607t^{6} + 17826923282143427t^{5} + 668953493325540t^{4} + 8043822999826t^{3} + 62795962475t^{2} + 226198975t + 65975 \right)D^{3} + 2 t ^{2} \left( 173653983780379639936t^{17} + 836353528712753683200t^{16} + 1859843639502479393664t^{15} + 2521948902248376130784t^{14} + 2331618276632270636576t^{13} + 1570615540608544081520t^{12} + 815084899795233160136t^{11} + 344265624731245651069t^{10} + 122970039215025736589t^{9} + 36840150001938337750t^{8} + 8723133958049159425t^{7} + 1497851710758605391t^{6} + 168414033906152748t^{5} + 10763872507420316t^{4} + 307593150782179t^{3} + 3323682532689t^{2} + 29836919115t + 93126952 \right)D^{2} + 4 t ^{2} \left( 94312077397964804448t^{17} + 441271560782054788752t^{16} + 952731134018165972960t^{15} + 1252704102355486005448t^{14} + 1121832292797201207602t^{13} + 732667375772013983599t^{12} + 370500111483452295555t^{11} + 153769269012067825357t^{10} + 54199788033622150800t^{9} + 15889405374444771670t^{8} + 3610535947717184542t^{7} + 580270829095544189t^{6} + 59217880110529206t^{5} + 3284087763285959t^{4} + 76095749753031t^{3} + 816303837118t^{2} + 6654575928t + 17626960 \right)D + 80 t ^{2} \left( 1924736273427853152t^{17} + 8826923529185180688t^{16} + 18667785573903875808t^{15} + 24014309833161003696t^{14} + 21018718645217098176t^{13} + 13424698270678393286t^{12} + 6664947917675904418t^{11} + 2732773535878588371t^{10} + 953625223725847035t^{9} + 274570806439957972t^{8} + 60293587839920971t^{7} + 9173578644366313t^{6} + 862820839133658t^{5} + 42300555435264t^{4} + 817373924611t^{3} + 8958444367t^{2} + 57841043t + 114920 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ6P6

|λ| = 31.02447 λ ≈ 31.02447; multiplicity 1; Jordan blocks 1 λ ≈ -12.14721 - 10.8525i; multiplicity 1; Jordan blocks 1 λ ≈ -12.14721 + 10.8525i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ -1.86502 - 0.1229397i; multiplicity 1; Jordan blocks 1 λ ≈ -1.86502 + 0.1229397i; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$31.0244694$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$31.02447$11
$-12.14721 - 10.8525\mathrm{i}$11
$-12.14721 + 10.8525\mathrm{i}$11
$-2$41, 3
$-1.86502 - 0.1229397\mathrm{i}$11
$-1.86502 + 0.1229397\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda^{5} - 3 \lambda^{4} - 510 \lambda^{3} - 10077 \lambda^{2} - 32411 \lambda - 28757\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{5} - 3 \lambda^{4} - 510 \lambda^{3} - 10077 \lambda^{2} - 32411 \lambda - 28757\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 28 & 396 & 192 & 540 & 3200 & 2112 & 13440 \\ 1 & -1 & 7 & 16 & 22 & 12 & 684 & 453 & 1536 \\ 1 & 1 & 0 & 72 & 48 & 82 & 651 & 393 & 1920 \\ 0 & 1 & 1 & -1 & -3 & 5 & 4 & 12 & 216 \\ 0 & 1 & 0 & 7 & 0 & 11 & 12 & 10 & 216 \\ 0 & 0 & 1 & 1 & 3 & -2 & 78 & 52 & 144 \\ 0 & 0 & 0 & 1 & -\frac{4}{13} & \frac{31}{13} & -5 & -1 & \frac{384}{13} \\ 0 & 0 & 0 & 1 & \frac{23}{13} & -\frac{6}{13} & 13 & 4 & -\frac{24}{13} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{11}{6} & \frac{7}{6} & 0\end{pmatrix}\notag\end{equation}$$