Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-82-24-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 6)
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_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w4) + E^(2w4) + E^(w2 + w3)
Bundle
O_2(1) + O_2(2) + Q_1 box U_2^v
ZL62 label
2051.3Aw6
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 82
$\chi(X, \mathrm{T}_X)$ = -38

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 82
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 0
$\mathrm{c}_4$ = 54
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 42 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 + 38 0 x0 + 38 x2
$\mathrm{HH}^{ 3 }$ = x1 + 9 0 0 x1 9
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 24 0 0 x1 - x2 - 1 1 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 42t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(642)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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) ^{3} \left( 269056t^{6} + 181248t^{5} + 38016t^{4} + 1760t^{3} + 24t - 1 \right) \left( 1793754243278110720t^{17} + 5333954324670709760t^{16} + 7273015053992968192t^{15} + 6163222985017192448t^{14} + 3691855734461632512t^{13} + 1672864624712411136t^{12} + 592232352285211392t^{11} + 164936422377443584t^{10} + 35780804191102912t^{9} + 5935814515224912t^{8} + 736998924303152t^{7} + 67049862021984t^{6} + 4391753074624t^{5} + 204545960860t^{4} + 6557164064t^{3} + 116995496t^{2} - 133866t + 533 \right)D^{6} + 2 \left( 4t + 1 \right) ^{2} \left( 60810163051608855092920320t^{25} + 245523610627287150745354240t^{24} + 465064636020608462850359296t^{23} + 554485307418400198995476480t^{22} + 470454624816096939604967424t^{21} + 303912629433248688311369728t^{20} + 155700822534355888828579840t^{19} + 64789072381091824316514304t^{18} + 22153487544301106242387968t^{17} + 6242567945170111142019072t^{16} + 1445478990686240462700544t^{15} + 273326884300306255224832t^{14} + 41873922233065112281600t^{13} + 5153630435289127646336t^{12} + 505498501523891037952t^{11} + 39308932878723218720t^{10} + 2434454448625586448t^{9} + 124337386505511064t^{8} + 5790669890600472t^{7} + 279151428227760t^{6} + 13406956530852t^{5} + 527615334628t^{4} + 14429867444t^{3} + 235423519t^{2} - 200266t + 533 \right)D^{5} + \left( 4t + 1 \right) \left( 4054010870107257006194688000t^{26} + 16755605865849967838494720000t^{25} + 32572512364163832300898877440t^{24} + 39988234629014381661895262208t^{23} + 35067740648263864700956049408t^{22} + 23509575743246521422317617152t^{21} + 12551707583935739272674410496t^{20} + 5466444364569638611650609152t^{19} + 1965586592703648086514597888t^{18} + 585713091244216456880455680t^{17} + 144410988860083130582368256t^{16} + 29326607873305832263995392t^{15} + 4875763795872784755425280t^{14} + 659154649248656110559232t^{13} + 71924986359480018108928t^{12} + 6280045975303004120448t^{11} + 433224722489016687264t^{10} + 22994733422430892816t^{9} + 871899531619979664t^{8} + 17386655303624400t^{7} - 334890458233736t^{6} - 39661858153836t^{5} - 1448117287132t^{4} - 30374732464t^{3} - 374078250t^{2} + 308487t - 533 \right)D^{4} + 2 t \left( 34053691308900958852035379200t^{26} + 145128004203291436902252544000t^{25} + 291628376286645364449670594560t^{24} + 371091337703140350400523141120t^{23} + 338270563438790121194963075072t^{22} + 236418876864509920818809536512t^{21} + 131995521448008472827478409216t^{20} + 60321676209140981692260417536t^{19} + 22853582614227242902714613760t^{18} + 7211889045456987699413909504t^{17} + 1895014150900191536095551488t^{16} + 413316934800993954278123520t^{15} + 74491583745392568178159616t^{14} + 11037459692625705022829568t^{13} + 1337312246814693166347648t^{12} + 131747018864000969466880t^{11} + 10485148875081658102176t^{10} + 667867658422058637696t^{9} + 33454587127565761136t^{8} + 1267982042694032040t^{7} + 33249546187172880t^{6} + 470537656864312t^{5} - 537505935424t^{4} - 121585703040t^{3} - 1842958152t^{2} + 2287197t + 1209 \right)D^{3} + 4 t ^{2} \left( 37621220874595345017486704640t^{25} + 156838937254443149120084377600t^{24} + 307993606710420755769662111744t^{23} + 383212762022507199505497063424t^{22} + 342089784794017659752143388672t^{21} + 234591558633283182636865683456t^{20} + 128712041034392640281137184768t^{19} + 57831942836874645071399895040t^{18} + 21521041973525263125039054848t^{17} + 6655565840292995415026343936t^{16} + 1708534896839268199887270912t^{15} + 362825644848134134505835520t^{14} + 63466058294967655217172736t^{13} + 9102875319081971803969472t^{12} + 1065610751984278651444800t^{11} + 101308342510814565089280t^{10} + 7771731560217632867360t^{9} + 475694997982166809208t^{8} + 22679302523751633408t^{7} + 797593405023484796t^{6} + 18174484688275216t^{5} + 172921228196266t^{4} - 1910096795026t^{3} - 42274068104t^{2} - 171301148t + 664569 \right)D^{2} + 32 t ^{2} \left( 5108053696335143827805306880t^{25} + 20947395082780228927986073600t^{24} + 40428937367826564327652982784t^{23} + 49458829679690518688911327232t^{22} + 43463804097011833773481951232t^{21} + 29385361133103282706253873152t^{20} + 15912679475901677272122187776t^{19} + 7057406495820215079119747072t^{18} + 2589485447956131407845292032t^{17} + 787915291810768112008637440t^{16} + 198460541679291227691528064t^{15} + 41234088767704854156354816t^{14} + 7038228466983325009701216t^{13} + 982956801899997860207624t^{12} + 111874156743786121889328t^{11} + 10330275293349000713680t^{10} + 768765506291774042912t^{9} + 45493651811599803080t^{8} + 2076209803554323630t^{7} + 68031581268231787t^{6} + 1328175037427595t^{5} + 5009982974767t^{4} - 318431465800t^{3} - 3663753961t^{2} + 3987833t + 30914 \right)D + 96 t ^{2} \left( 694973292018386915347660800t^{25} + 2818039931662480646864896000t^{24} + 5374100354208420280349491200t^{23} + 6497665829881695589816598528t^{22} + 5648132732524405457542774784t^{21} + 3781080758415599543901749248t^{20} + 2028792391022147461134434304t^{19} + 891491538905201112133234688t^{18} + 323766168463466772793643008t^{17} + 97337784416886635725877248t^{16} + 24171983784930886435942144t^{15} + 4940200309540551856393728t^{14} + 827739712561889539533888t^{13} + 113281685556470081514896t^{12} + 12618361438275429526464t^{11} + 1139306639086535243560t^{10} + 82811912786700918032t^{9} + 4772931379951654196t^{8} + 210491584919774660t^{7} + 6527648686771864t^{6} + 112265280884868t^{5} - 96731577224t^{4} - 38125316560t^{3} - 389388745t^{2} + 162071t + 2132 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 6)

|λ| = 28.20728 λ ≈ 28.20728; multiplicity 1; Jordan blocks 1 λ ≈ 3.71362 - 12.09449i; multiplicity 1; Jordan blocks 1 λ ≈ 3.71362 + 12.09449i; multiplicity 1; Jordan blocks 1 λ ≈ -5.088226; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 3; Jordan blocks 3 λ ≈ -3.273146 - 0.9989892i; multiplicity 1; Jordan blocks 1 λ ≈ -3.273146 + 0.9989892i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 3
Sector dimension
10
Spectral radius
$28.207279$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28.20728$11
$3.71362 - 12.09449\mathrm{i}$11
$3.71362 + 12.09449\mathrm{i}$11
$-5.088226$11
$-4$33
$-3.273146 - 0.9989892\mathrm{i}$11
$-3.273146 + 0.9989892\mathrm{i}$11
$-3$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{3} \left(\lambda^{6} - 24 \lambda^{5} - 1760 \lambda^{3} - 38016 \lambda^{2} - 181248 \lambda - 269056\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{3} \left(\lambda^{6} - 24 \lambda^{5} - 1760 \lambda^{3} - 38016 \lambda^{2} - 181248 \lambda - 269056\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 44 & 168 & 84 & 180 & 288 & 1376 & 2048 & 8000 \\ 1 & 0 & 6 & 28 & -8 & 40 & 52 & 36 & 12 & 64 \\ 1 & 0 & 4 & 16 & 12 & 24 & 36 & 192 & 300 & 1344 \\ 0 & 1 & 1 & 4 & -1 & 6 & 10 & 8 & 0 & -48 \\ 0 & 1 & 0 & 6 & 1 & 2 & 6 & 44 & 64 & 192 \\ 0 & 0 & 1 & 0 & 1 & -2 & 2 & 4 & 8 & 72 \\ 0 & 0 & 1 & 0 & 0 & 2 & -2 & 12 & 20 & 96 \\ 0 & 0 & 0 & -1 & -2 & 4 & 3 & -14 & -18 & -64 \\ 0 & 0 & 0 & 3 & 2 & -1 & 1 & 14 & 18 & 72 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 2 & 0\end{pmatrix}\notag\end{equation}$$