Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-146-36-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3 + w5) + E^(w2 + w5) + E^(w1 + w5)
Bundle
O_3(1) box O_4(1) + O_2(1) box O_4(1) + O_1(1) box O_4(1)
ZL62 label
11131.qnj
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 146
$\chi(X, \mathrm{T}_X)$ = -9

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 146
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 110
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 14 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 + 9 0 x0 + 9 x1
$\mathrm{HH}^{ 3 }$ = x1 + 22 0 0 x1 + 9 13
$\mathrm{HH}^{ 4 }$ = 36 0 0 0 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 82t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(542)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(579)

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( 4t + 1 \right) \left( 5t - 1 \right) \left( 3t + 1 \right) ^{2} \left( 40t^{2} + 12t + 1 \right) \left( 72t^{2} + 12t - 1 \right) \left( 28844400015360t^{11} + 49751357709312t^{10} + 31762759590528t^{9} + 7944159219328t^{8} - 540160907248t^{7} - 814007998800t^{6} - 206311382188t^{5} - 24620703740t^{4} - 1347130530t^{3} - 16266800t^{2} + 763925t - 5037 \right)D^{6} + \left( 3t + 1 \right) \left( 209341117551476736000t^{19} + 600403708781730201600t^{18} + 733931503221175418880t^{17} + 487074361361396908032t^{16} + 176393484581283053568t^{15} + 21238252398441007104t^{14} - 10519445573448933376t^{13} - 5987674555587316480t^{12} - 1410824079082477568t^{11} - 158122014080852384t^{10} + 1398768412133104t^{9} + 3137392640918856t^{8} + 481740192961244t^{7} + 40648697822396t^{6} + 2512969258458t^{5} + 142305464526t^{4} + 6012289461t^{3} + 78569518t^{2} - 2362293t + 10074 \right)D^{5} + \left( 5233527938786918400000t^{20} + 15888956959994904576000t^{19} + 20861840023975150387200t^{18} + 15238148049713971814400t^{17} + 6422995221303388428288t^{16} + 1235449030568726075904t^{15} - 204268815681029464320t^{14} - 212916576297900720320t^{13} - 69218798899417102848t^{12} - 12875431439880750384t^{11} - 1379576715240103456t^{10} - 53843047980341576t^{9} + 6317058339765444t^{8} + 1030133721542060t^{7} + 51385211564978t^{6} - 774560238360t^{5} - 214487681959t^{4} - 9489075536t^{3} - 116937090t^{2} + 1231012t - 5037 \right)D^{4} + t \left( 21980817342905057280000t^{19} + 63876990492803432448000t^{18} + 79713969035947484774400t^{17} + 54705896587265750876160t^{16} + 21028949829899240509440t^{15} + 3069326047403161331712t^{14} - 1162368094386449069568t^{13} - 823223908808461540352t^{12} - 245880050638770601600t^{11} - 44628369564889298880t^{10} - 5091968064654558192t^{9} - 327173837941391752t^{8} - 4897777107877228t^{7} + 870529504783392t^{6} + 58371369940366t^{5} + 687173257012t^{4} - 30991721561t^{3} + 305684149t^{2} + 30255773t + 9729 \right)D^{3} + 2 t ^{2} \left( 24283569635971301376000t^{18} + 68154334252984782028800t^{17} + 81575291850280425308160t^{16} + 53069966167433955176448t^{15} + 18713660187596540470272t^{14} + 1876573739774852718336t^{13} - 1461354798392008486272t^{12} - 840180134922715832352t^{11} - 234077375986523089536t^{10} - 40918998151635465944t^{9} - 4641317114477680968t^{8} - 322246387456985296t^{7} - 10439007702083440t^{6} + 159481012066848t^{5} + 22000062974872t^{4} + 300463560328t^{3} - 3463743969t^{2} + 564215033t + 9894420 \right)D^{2} + 8 t ^{2} \left( 6594245202871517184000t^{18} + 18026937990214542950400t^{17} + 20887043122124905144320t^{16} + 13012766729211251294208t^{15} + 4253257267843484533248t^{14} + 247455161229331798272t^{13} - 420137321703187419072t^{12} - 212702903913339003456t^{11} - 56129726389581942512t^{10} - 9460863992430747092t^{9} - 1049299479863283356t^{8} - 73279623231252672t^{7} - 2694391812618958t^{6} - 7126649532190t^{5} + 2837212551879t^{4} + 54212474791t^{3} + 720387106t^{2} + 105560026t + 876438 \right)D + 48 t ^{2} \left( 448588109038878720000t^{18} + 1204239529798336512000t^{17} + 1363499555024074752000t^{16} + 822844049897846937600t^{15} + 253245355910696484864t^{14} + 5890969246798574208t^{13} - 29312051506470617856t^{12} - 13712255183119441296t^{11} - 3479093922246896400t^{10} - 569307103100098384t^{9} - 61677113575367688t^{8} - 4250931637956976t^{7} - 159798816572696t^{6} - 1118629877398t^{5} + 129673197298t^{4} + 3492541092t^{3} + 83581277t^{2} + 4844789t + 23506 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × Gr(2, 4)

|λ| = 16.3923 λ ≈ 16.3923; multiplicity 1; Jordan blocks 1 λ ≈ -6 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -6 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ 5; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -4.392305; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -3; multiplicity 6; Jordan blocks 1, 1, 1, 1, 1, 1 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 3 3 6 2
Sector dimension
18
Spectral radius
$16.3923048$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$16.3923$11
$-6 - 2\mathrm{i}$11
$-6 + 2\mathrm{i}$11
$5$31, 1, 1
$-4.392305$11
$-4$31, 1, 1
$-3$61, 1, 1, 1, 1, 1
$-2$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{6} \left(\lambda + 4\right)^{3} \left(\lambda - 5\right)^{3} \left(\lambda^{2} + 12 \lambda + 40\right) \left(\lambda^{2} - 12 \lambda - 72\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda + 4\right) \left(\lambda - 5\right) \left(\lambda^{2} + 12 \lambda + 40\right) \left(\lambda^{2} - 12 \lambda - 72\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 4 & 16 & 6 & 6 & 24 & 6 & 24 & 24 & 27 & 27 & 8 & 40 & 40 & 40 & 80 \\ 1 & -1 & 0 & 0 & 2 & 4 & 4 & 12 & 0 & 4 & 4 & 6 & 6 & 6 & 18 & 18 & 6 & 32 \\ 1 & 0 & -1 & 0 & 2 & 4 & 0 & 4 & 4 & 12 & 4 & 6 & 6 & 6 & 18 & 6 & 18 & 32 \\ 1 & 0 & 0 & -1 & 2 & 0 & 4 & 4 & 4 & 4 & 12 & 6 & 6 & 6 & 6 & 18 & 18 & 32 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 4 & 0 & 4 & 4 & 6 & 6 & 0 & 6 & 6 & 6 & 8 \\ 0 & 1 & 1 & 0 & 0 & -2 & 0 & 2 & 0 & 2 & 0 & 1 & 1 & 4 & 8 & 4 & 4 & 12 \\ 0 & 1 & 0 & 1 & 0 & 0 & -2 & 2 & 0 & 0 & 2 & 1 & 1 & 4 & 4 & 8 & 4 & 12 \\ 0 & 1 & 0 & 0 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 1 & 1 & 0 & 4 & 4 & 0 & 6 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -2 & 2 & 2 & 1 & 1 & 4 & 4 & 4 & 8 & 12 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & 1 & 0 & 4 & 0 & 4 & 6 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & -1 & 1 & 1 & 0 & 0 & 4 & 4 & 6 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & 1 & -1 & -1 & -\frac{5}{2} & -\frac{5}{2} & -3 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 0 & \frac{3}{2} & \frac{3}{2} & 1 & 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 2 & \frac{3}{2} & \frac{3}{2} & 1 & 1 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & \frac{3}{2} & \frac{3}{2} & 1 & 1 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$