Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-60-19-a

·

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

Defining data
Ambient
ℙ3P3 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w5) + E^(w1 + w5) + E^(2w1 + w5)
Bundle
O_2(1) + O_1(1) box O_2(1) + O_1(2) box O_2(1)
ZL62 label
3031.4jx337
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 19
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 60
$\chi(X, \mathrm{T}_X)$ = -39

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 60
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 156
$\mathrm{c}_1\mathrm{c}_3$ = -4
$\mathrm{c}_4$ = 68
Hodge diamond
1
00
0 2 0
0 0 0 0
0 5 52 5 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 + 39 0 x0 + 39 x1
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 + 5 1
$\mathrm{HH}^{ 4 }$ = 34 0 0 15 0 19
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 14t + 30t^{2} + 14t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(677)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(677)

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( 4t + 1 \right) ^{4} \left( 18032t^{4} + 7392t^{3} + 760t^{2} + 12t - 1 \right) \left( 36263136698544t^{10} + 70095019702624t^{9} + 45677660047964t^{8} + 13508609472784t^{7} + 1714873827204t^{6} - 7228686864t^{5} - 24953405649t^{4} - 2252911964t^{3} - 46601284t^{2} + 779896t - 1380 \right)D^{6} + 2 \left( 4t + 1 \right) ^{3} \left( 22232493952236943872t^{15} + 53779600760260634112t^{14} + 49463013503626377088t^{13} + 23617693531643622208t^{12} + 6376437226584315744t^{11} + 915083462447989536t^{10} + 31957687616269016t^{9} - 11411025998930596t^{8} - 2055894649934096t^{7} - 150332715518288t^{6} - 4832028165678t^{5} - 64032724203t^{4} - 3140468542t^{3} - 107764312t^{2} + 1183644t - 1380 \right)D^{5} + \left( 4t + 1 \right) ^{2} \left( 1203170260944587550720t^{16} + 3053350849612778926400t^{15} + 2972141275065451288576t^{14} + 1520735409865426991824t^{13} + 448694332279642023008t^{12} + 73470305785150441312t^{11} + 4063075389067012960t^{10} - 824513063703130252t^{9} - 201636970976301068t^{8} - 19569923990574776t^{7} - 947168290707184t^{6} - 17707411482590t^{5} + 238811495101t^{4} + 16774180344t^{3} + 238225068t^{2} - 1331448t + 1380 \right)D^{4} + 4 t \left( 4t + 1 \right) \left( 4132628287592278978560t^{16} + 11121219035373111719040t^{15} + 11668977574963829209824t^{14} + 6546856754368879805472t^{13} + 2171613968163825904824t^{12} + 421718980205849009904t^{11} + 37026569261216431104t^{10} - 2729511733282912088t^{9} - 1208598882345114734t^{8} - 156139398855956928t^{7} - 10417785209872548t^{6} - 338645204119208t^{5} - 2436267623186t^{4} + 100649533737t^{3} + 546801429t^{2} - 11949566t - 1886 \right)D^{3} + 2 t ^{2} \left( 60597931751226531250176t^{16} + 174019091744043680931328t^{15} + 198776930921082218523392t^{14} + 123639685897261206219680t^{13} + 46594072584603007091264t^{12} + 10778860204928799631560t^{11} + 1346015878712246053104t^{10} + 7693852602515477936t^{9} - 28533645507221550960t^{8} - 5240205189986044998t^{7} - 482963487305682208t^{6} - 24415011603007932t^{5} - 565284902223228t^{4} + 261502497655t^{3} + 170552304934t^{2} - 979307078t - 13336500 \right)D^{2} + 16 t ^{2} \left( 6988849863573778120704t^{16} + 19731376132496602422912t^{15} + 21874707755149903239040t^{14} + 13077206187169203545872t^{13} + 4678841319750411688224t^{12} + 1003148562025716106092t^{11} + 106192295781690446456t^{10} - 4044188596052695260t^{9} - 3144071388323323144t^{8} - 489147540025542283t^{7} - 39967520775881576t^{6} - 1759667501217253t^{5} - 32075639111780t^{4} + 202696012522t^{3} + 9161128484t^{2} - 112441320t - 466440 \right)D + 16 t ^{2} \left( 2510964022840878366720t^{16} + 7007087043349533826560t^{15} + 7604243995877762072448t^{14} + 4415959361814782187728t^{13} + 1519115242786012721024t^{12} + 306430422401716484612t^{11} + 27709647643146676560t^{10} - 2368898453205635196t^{9} - 1069305734136077416t^{8} - 148915753461807073t^{7} - 11080155910544574t^{6} - 433849770786075t^{5} - 6236067465352t^{4} + 75317962088t^{3} + 1420860624t^{2} - 28625708t - 60720 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 4)

|λ| = 37.69814 λ ≈ 37.69814; multiplicity 1; Jordan blocks 1 λ ≈ -10.93844 - 2.350643i; multiplicity 1; Jordan blocks 1 λ ≈ -10.93844 + 2.350643i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 λ ≈ -3.821256; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$37.6981426$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$37.69814$11
$-10.93844 - 2.350643\mathrm{i}$11
$-10.93844 + 2.350643\mathrm{i}$11
$-4$41, 3
$-3.821256$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right)^{4} \left(\lambda^{4} - 12 \lambda^{3} - 760 \lambda^{2} - 7392 \lambda - 18032\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right)^{3} \left(\lambda^{4} - 12 \lambda^{3} - 760 \lambda^{2} - 7392 \lambda - 18032\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 44 & 44 & 636 & 384 & 240 & 3296 & 5408 & 10640 \\ 1 & 0 & 6 & 76 & 52 & 22 & 390 & 624 & 1280 \\ 1 & 5 & 0 & 88 & 50 & 32 & 402 & 660 & 1376 \\ 0 & 1 & 1 & 0 & 3 & 3 & 44 & 76 & 132 \\ 0 & 1 & 0 & 6 & -2 & 2 & 14 & 28 & 72 \\ 0 & 0 & 2 & 10 & 8 & -2 & 56 & 84 & 168 \\ 0 & 0 & 0 & 2 & -\frac{2}{7} & \frac{13}{7} & \frac{12}{7} & \frac{78}{7} & \frac{176}{7} \\ 0 & 0 & 0 & 1 & \frac{10}{7} & -\frac{2}{7} & \frac{17}{7} & -\frac{12}{7} & \frac{44}{7} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 4 & 0\end{pmatrix}\notag\end{equation}$$