Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 5-200-46-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 4) × 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_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7 (marked)node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w7) + E^(w4) + E^(w2 + w7) + E^(w2 + w4)
Bundle
O_3(1) + O_2(1) + Q_1 box O_3(1) + Q_1 box O_2(1)
ZL62 label
203131.63wBw8
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 46
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 200
$\chi(X, \mathrm{T}_X)$ = 0

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 200
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 24
Hodge diamond
1
00
0 5 0
0 0 0 0
0 0 12 0 0
0 0 0 0
0 5 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 0 x0 x1
$\mathrm{HH}^{ 3 }$ = x1 + 23 0 0 x1 - 7 30
$\mathrm{HH}^{ 4 }$ = 46 0 0 0 0 46
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 41t + 116t^{2} + 41t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(237)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(308)

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( 4t + 1 \right) ^{2} \left( 16t^{2} + 1 \right) \left( 496t^{3} - 48t^{2} + 12t - 1 \right) \left( 249141832187904t^{13} + 199059371384832t^{12} + 127042278592512t^{11} - 22973494273024t^{10} - 37489324341760t^{9} - 12077364901664t^{8} - 1560354205168t^{7} - 106844388096t^{6} - 20340970460t^{5} - 4140778066t^{4} - 335900902t^{3} - 11139263t^{2} - 54612t - 2200 \right)D^{6} + 2 \left( 4t + 1 \right) \left( 201673337184807026688t^{20} + 233785794605705330688t^{19} + 179342449469797957632t^{18} + 26142482893045235712t^{17} - 37539062065884299264t^{16} - 25844999947465195520t^{15} - 8677699137713913856t^{14} - 1962137915737231360t^{13} - 346966393636290560t^{12} - 48912838506065024t^{11} - 6315978578759808t^{10} - 1007869835923136t^{9} - 171872374156488t^{8} - 18990981338008t^{7} - 1089792537646t^{6} - 61811665232t^{5} - 10017007919t^{4} - 759393655t^{3} - 22189360t^{2} - 75318t - 2200 \right)D^{5} + \left( 10914086482942497914880t^{21} + 14325444395220256948224t^{20} + 11718420206789597331456t^{19} + 2419595839397432918016t^{18} - 2494463655283765018624t^{17} - 2072468446903538778112t^{16} - 758322120861091610624t^{15} - 171965128627651885056t^{14} - 28803675209287523328t^{13} - 4330533521351081088t^{12} - 678097245570448576t^{11} - 105634676734680192t^{10} - 13912389540937616t^{9} - 1322754256829608t^{8} - 86266876894728t^{7} - 3559399334396t^{6} + 86995037074t^{5} + 28671346070t^{4} + 1789588469t^{3} + 45583739t^{2} + 188336t + 2200 \right)D^{4} + 2 t \left( 18743757220705594245120t^{20} + 23236987174501503467520t^{19} + 18767400920221483008000t^{18} + 2360003577927415627776t^{17} - 5212017498598007767040t^{16} - 3662161299575978999808t^{15} - 1190426294933008195584t^{14} - 234765836133812032512t^{13} - 34903548336950359552t^{12} - 5514423888828098752t^{11} - 1006977833213682304t^{10} - 164874493797564416t^{9} - 21080039195505200t^{8} - 1983124489736956t^{7} - 128039844146120t^{6} - 5240717109794t^{5} - 134129040066t^{4} - 3224600422t^{3} - 72455371t^{2} - 1300067t - 1056 \right)D^{3} + 8 t ^{2} \left( 8588911536576487489536t^{19} + 10183728304587936890880t^{18} + 8169904841741423345664t^{17} + 471532705202515673088t^{16} - 2679336376388863066112t^{15} - 1692810513307972599808t^{14} - 503681228471053936640t^{13} - 89694548393767275776t^{12} - 12723159883014300160t^{11} - 2280166122077577184t^{10} - 457191030147894344t^{9} - 71235996197227712t^{8} - 7964900798003090t^{7} - 644706904496866t^{6} - 37225516080799t^{5} - 1382587077360t^{4} - 32757969910t^{3} - 1006633057t^{2} - 22765042t - 258250 \right)D^{2} + 16 t ^{2} \left( 3962287918807385112576t^{19} + 4541982672772682219520t^{18} + 3632035364236602114048t^{17} + 10110023721330769920t^{16} - 1330154034643639042048t^{15} - 782605402560465129472t^{14} - 217990828265282675712t^{13} - 36104687386997878400t^{12} - 5136083100786089728t^{11} - 1042855672006309120t^{10} - 215887001792216816t^{9} - 31623843499851632t^{8} - 3162799275557602t^{7} - 225946924200568t^{6} - 12138078261085t^{5} - 433585857427t^{4} - 9374083284t^{3} - 307725004t^{2} - 6124656t - 52800 \right)D + 32 t ^{2} \left( 711788248887554211840t^{19} + 796197775178956013568t^{18} + 635795361282125463552t^{17} - 24779941538773303296t^{16} - 250687294115866312704t^{15} - 140624911705465171968t^{14} - 37352093104398289920t^{13} - 5888919078260264832t^{12} - 861009396264809984t^{11} - 192081368763960384t^{10} - 39874214791437736t^{9} - 5539510540595120t^{8} - 508736877589296t^{7} - 32887481327712t^{6} - 1693245822848t^{5} - 61150513998t^{4} - 1273744065t^{3} - 41073990t^{2} - 655512t - 4400 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 4) × Gr(2, 4)

|λ| = 11.55953 λ ≈ 11.55953; multiplicity 1; Jordan blocks 1 λ ≈ 0.2202369 - 6.546742i; multiplicity 1; Jordan blocks 1 λ ≈ 0.2202369 + 6.546742i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ 0 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2 2 2 2
Sector dimension
12
Spectral radius
$11.5595263$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.55953$11
$0.2202369 - 6.546742\mathrm{i}$11
$0.2202369 + 6.546742\mathrm{i}$11
$-4$22
$0 - 4\mathrm{i}$21, 1
$0 + 4\mathrm{i}$21, 1
$-3$11
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 3\right) \left(\lambda + 4\right)^{2} \left(\lambda^{2} + 16\right)^{2} \left(\lambda^{3} - 12 \lambda^{2} + 48 \lambda - 496\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 3\right) \left(\lambda + 4\right)^{2} \left(\lambda^{2} + 16\right) \left(\lambda^{3} - 12 \lambda^{2} + 48 \lambda - 496\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 4 & 12 & 12 & 12 & 12 & 16 & 64 & 16 & 40 \\ 1 & 0 & 2 & 2 & 0 & 0 & -8 & 0 & -6 & 0 & -6 & 0 \\ 1 & 0 & 0 & 0 & 0 & 4 & 4 & 4 & 6 & 12 & 6 & 16 \\ 1 & 0 & 0 & 0 & 4 & 0 & 4 & 4 & 6 & 12 & 6 & 16 \\ 0 & 1 & 2 & 0 & 0 & 0 & -1 & 2 & -2 & 4 & 0 & 0 \\ 0 & 1 & 0 & 2 & 0 & 0 & -1 & 2 & 0 & 4 & -2 & 0 \\ 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 6 \\ 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 & 0 & 4 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0 & 0 & 4 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 & 1 & 0\end{pmatrix}\notag\end{equation}$$