Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-84-24-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ7P7
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_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(2w4) + 2 copies of E^(w1 + w4) + E^(w3 + w4)
Bundle
O_2(2) + 2 copies of O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
30700.DwHw2w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;8) \\ \det\mathcal{E} &\cong \mathcal{O}(3;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 84
$\chi(X, \mathrm{T}_X)$ = -34

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 84
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 140
$\mathrm{c}_1\mathrm{c}_3$ = 4
$\mathrm{c}_4$ = 52
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 40 3 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 + 34 0 x0 + 34 x2
$\mathrm{HH}^{ 3 }$ = x1 + 8 0 0 x1 8
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 25 0 0 x1 - x2 + 1 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 44t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(606)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(624)

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( 25t - 1 \right) \left( 2t + 1 \right) ^{5} \left( 158t^{2} + 17t + 1 \right) \left( 7628677760t^{7} + 10814597792t^{6} + 5598851216t^{5} + 1241999592t^{4} + 129464952t^{3} + 16657668t^{2} + 121801t + 483 \right)D^{6} + \left( 2t + 1 \right) ^{4} \left( 3796792921152000t^{12} + 7503820942224640t^{11} + 6093293336521152t^{10} + 2606476845968832t^{9} + 642607675985040t^{8} + 101911577845152t^{7} + 12311422353300t^{6} + 990086775484t^{5} + 32941144029t^{4} + 538231455t^{3} + 65480653t^{2} + 358641t + 966 \right)D^{5} + \left( 2t + 1 \right) ^{3} \left( 63279882019200000t^{13} + 139380390619801600t^{12} + 128857049911830656t^{11} + 64612927626575552t^{10} + 19275712395771616t^{9} + 3704816887839792t^{8} + 521277011164168t^{7} + 54858622935996t^{6} + 3071039340962t^{5} + 46024694947t^{4} + 122547392t^{3} - 59773092t^{2} - 306162t - 483 \right)D^{4} + t \left( 2t + 1 \right) ^{2} \left( 531551008961280000t^{13} + 1322237449469593600t^{12} + 1405060153503020800t^{11} + 829210017774214400t^{10} + 298827191118353536t^{9} + 69972395012015040t^{8} + 11607632025069792t^{7} + 1473572446599456t^{6} + 127262145454624t^{5} + 5206215281108t^{4} + 69145883447t^{3} + 644210511t^{2} + 3262879t + 1219 \right)D^{3} + 2 \left( 2t + 1 \right) t ^{2} \left( 1174474610276352000t^{13} + 3315474648932088320t^{12} + 4057895243861900544t^{11} + 2812815885080886144t^{10} + 1217331835891769536t^{9} + 347081394859208480t^{8} + 69045655315962896t^{7} + 10306879437305544t^{6} + 1155379611181460t^{5} + 80380223277900t^{4} + 2453193523630t^{3} + 28713665790t^{2} + 331410719t + 1341963 \right)D^{2} + 16 t ^{2} \left( 318930605376768000t^{14} + 1021326103067426560t^{13} + 1436501003324983808t^{12} + 1163499868807673024t^{11} + 599854676791141824t^{10} + 207086915974506352t^{9} + 49867088382668896t^{8} + 8824132588791860t^{7} + 1196204185606372t^{6} + 116973852632338t^{5} + 6638677971334t^{4} + 158537455658t^{3} + 1836132626t^{2} + 19448927t + 64239 \right)D + 96 t ^{2} \left( 21695959549440000t^{14} + 67710289382796800t^{13} + 92537945780303872t^{12} + 72550825712748352t^{11} + 36048637430796224t^{10} + 11952794548275840t^{9} + 2767016831210496t^{8} + 474297656056572t^{7} + 62387766770720t^{6} + 5750284159334t^{5} + 289214024369t^{4} + 5794037508t^{3} + 69751122t^{2} + 589398t + 1449 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ7P7

|λ| = 25 λ ≈ 25; multiplicity 1; Jordan blocks 1 λ ≈ -8.5 - 9.26013i; multiplicity 1; Jordan blocks 1 λ ≈ -8.5 + 9.26013i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 5; Jordan blocks 1, 1, 3 5
Sector dimension
9
Spectral radius
$25$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$25$11
$-8.5 - 9.26013\mathrm{i}$11
$-8.5 + 9.26013\mathrm{i}$11
$-3$11
$-2$51, 1, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{5} \left(\lambda + 3\right) \left(\lambda - 25\right) \left(\lambda^{2} + 17 \lambda + 158\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda + 3\right) \left(\lambda - 25\right) \left(\lambda^{2} + 17 \lambda + 158\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 20 & 228 & 108 & 312 & 1632 & 944 & 5760 \\ 1 & -1 & 6 & 8 & 8 & 12 & 312 & 204 & 896 \\ 1 & 1 & 0 & 44 & 28 & 48 & 372 & 204 & 960 \\ 0 & 1 & 1 & -1 & -2 & 6 & 4 & 8 & 144 \\ 0 & 1 & 0 & 6 & 0 & 8 & 12 & 4 & 96 \\ 0 & 0 & 1 & 1 & 2 & -2 & 44 & 28 & 96 \\ 0 & 0 & 0 & 1 & -\frac{1}{5} & \frac{13}{5} & -3 & 0 & \frac{112}{5} \\ 0 & 0 & 0 & 1 & \frac{8}{5} & -\frac{4}{5} & 10 & 2 & -\frac{16}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{4} & 1 & 0\end{pmatrix}\notag\end{equation}$$