Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-256-57-l

·

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

Defining data
Ambient
Fl(1, 2, 3) × ℙ3P3
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_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w3) + E^(w2 + w3)
Bundle
O_1(1,0) box O_2(1) + O_1(0,1) box O_2(1)
ZL62 label
2230.da
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1,1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 57
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 256
$\chi(X, \mathrm{T}_X)$ = 1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 256
$\mathrm{c}_1^2\mathrm{c}_2$ = 160
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 56
$\mathrm{c}_4$ = 14
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 6 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 1 0 x0 - 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 43 0 0 x1 - 14 57
$\mathrm{HH}^{ 4 }$ = 57 0 0 0 0 57
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 52t + 150t^{2} + 52t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(193)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(269)

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( 16t^{2} + 8t - 1 \right) \left( 32t^{2} + 1 \right) \left( 37t^{4} + 24t^{3} + 10t^{2} + 7t + 1 \right) \left( 6568447163904t^{13} + 12843045511552t^{12} - 1905909034848t^{11} - 24241317987328t^{10} - 26698255405592t^{9} - 14651073993888t^{8} - 4998273112452t^{7} - 1238473866484t^{6} - 231850448012t^{5} - 26644778982t^{4} - 1629414268t^{3} - 99504684t^{2} - 311393t - 12352 \right)D^{6} + \left( 5226171849065889792t^{22} + 17680096005500698624t^{21} + 16459891537409064960t^{20} - 15207772304934739968t^{19} - 53529005131184015360t^{18} - 65811197311430510592t^{17} - 51516069524918131904t^{16} - 29859616991767123808t^{15} - 13701789862859488848t^{14} - 5088903028326903720t^{13} - 1533235271775775656t^{12} - 375980802471451444t^{11} - 75008800471510400t^{10} - 11924647324473604t^{9} - 1498147964971760t^{8} - 164423902958496t^{7} - 18559490096578t^{6} - 2052930820431t^{5} - 175430080004t^{4} - 8229009420t^{3} - 400078927t^{2} - 909475t - 24704 \right)D^{5} + \left( 43551432075549081600t^{22} + 139637009204221050880t^{21} + 106448209801181704192t^{20} - 173585086322344813568t^{19} - 472312999108321236736t^{18} - 530055568729179195776t^{17} - 380594686919879452672t^{16} - 199976281340457311616t^{15} - 82537634001935386672t^{14} - 27694399481597258120t^{13} - 7644171077383619032t^{12} - 1741218875401924508t^{11} - 321787279494259804t^{10} - 45344826872995304t^{9} - 4423857481146834t^{8} - 262740115091048t^{7} - 2181617374018t^{6} + 1978270105683t^{5} + 203592154338t^{4} + 7841751572t^{3} + 364025296t^{2} + 690915t + 12352 \right)D^{4} + t \left( 182916014717306142720t^{21} + 561074928880388669440t^{20} + 349189289036191211520t^{19} - 865991956786138118144t^{18} - 2037852390604047923712t^{17} - 2143267912398077367808t^{16} - 1447873747161105891008t^{15} - 711784184295876404000t^{14} - 273486851384575808624t^{13} - 85272338370872905656t^{12} - 21880942646841958600t^{11} - 4648340306223409980t^{10} - 808269791777052568t^{9} - 109374807982608748t^{8} - 10895566875918888t^{7} - 808386554533696t^{6} - 44287648156638t^{5} - 1139408564085t^{4} - 26089268272t^{3} - 339436974t^{2} - 12269450t - 6369 \right)D^{3} + 2 t ^{2} \left( 202078644830547738624t^{20} + 598384728641773174784t^{19} + 305182303166145996800t^{18} - 1062205369647549943296t^{17} - 2280179598561916137088t^{16} - 2289121143876197852864t^{15} - 1481839847542420470560t^{14} - 697438179393509923376t^{13} - 256182622879133063616t^{12} - 76044575169617915992t^{11} - 18405527334858233600t^{10} - 3652807201403542992t^{9} - 593028088541066680t^{8} - 75685927049889932t^{7} - 7233570252781376t^{6} - 524802269997858t^{5} - 26086847142890t^{4} - 423429528885t^{3} - 8236521262t^{2} - 259719637t - 4982400 \right)D^{2} + 4 t ^{2} \left( 109749608830383685632t^{20} + 316440006368985677824t^{19} + 134265832618413864960t^{18} - 616170620518254790144t^{17} - 1245715292738301194368t^{16} - 1210332404106962532096t^{15} - 760892241377959450272t^{14} - 348278317091001050240t^{13} - 124489480838034298320t^{12} - 35792581654380985752t^{11} - 8283757107641870552t^{10} - 1548601760562366444t^{9} - 235778838512159104t^{8} - 28527035118701880t^{7} - 2630945255249364t^{6} - 187785502799654t^{5} - 8606040029358t^{4} - 85533781605t^{3} - 1606097900t^{2} - 71392816t - 1086976 \right)D + 8 t ^{2} \left( 22397879353139527680t^{20} + 63401461885259612160t^{19} + 23110818956738482176t^{18} - 130944015240337876992t^{17} - 254915632643837128192t^{16} - 242369290747805933376t^{15} - 149486731864352529984t^{14} - 67256180431489285424t^{13} - 23658106600589957728t^{12} - 6667149680794858040t^{11} - 1494592693488428816t^{10} - 266679098607239288t^{9} - 38514637739137416t^{8} - 4456767005757200t^{7} - 398557322706184t^{6} - 28037230042800t^{5} - 1208866824702t^{4} - 8416149395t^{3} - 194428364t^{2} - 7313156t - 98816 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3) × ℙ3P3

|λ| = 9.656854 λ ≈ 9.656854; multiplicity 1; Jordan blocks 1 λ ≈ -5.799605; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ 0.2345522 - 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 0.2345522 + 1.940705i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.6695; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.656854; multiplicity 1; Jordan blocks 1 2 2 2 2
Sector dimension
13
Spectral radius
$9.65685425$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$9.656854$11
$-5.799605$21, 1
$0 - 5.656854\mathrm{i}$11
$0 + 5.656854\mathrm{i}$11
$-2$11
$0.2345522 - 1.940705\mathrm{i}$21, 1
$0.2345522 + 1.940705\mathrm{i}$21, 1
$-1.6695$21, 1
$-1.656854$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{2} + 32\right) \left(\lambda^{2} - 8 \lambda - 16\right) \left(\lambda^{4} + 7 \lambda^{3} + 10 \lambda^{2} + 24 \lambda + 37\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{2} + 32\right) \left(\lambda^{2} - 8 \lambda - 16\right) \left(\lambda^{4} + 7 \lambda^{3} + 10 \lambda^{2} + 24 \lambda + 37\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 2 & 0 & 3 & 0 & 3 & 6 & 4 & 8 & 16 & 10 \\ 1 & -1 & 1 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 3 & 3 & 4 \\ 1 & 1 & -1 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 6 & 4 \\ 2 & 1 & 1 & 0 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 4 \\ 0 & 1 & 2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 3 \\ 0 & 2 & 0 & 1 & 1 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 4 & 3 \\ 0 & 0 & 2 & 1 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 1 & 0 & 1 & 0 & 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 2 & 3 & -1 & -3 & -1 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & -1 & \frac{9}{2} & \frac{5}{2} & \frac{1}{2} & -1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 & 1 & \frac{3}{2} & \frac{5}{2} & \frac{1}{2} & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 5 & 0\end{pmatrix}\notag\end{equation}$$