Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-84-24-a

·

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

Defining data
Ambient
Fl(1, 5, 7)
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
2 copies of E^(w5) + 3 copies of E^(w6) + 2 copies of E^(2w1)
Bundle
2 copies of O(0,1) + 3 copies of Q_1^v ot wedge^4 U_2^v ot U_3^v + 2 copies of O(2,0)
ZL62 label
60G.5554466
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5,6) \\ \det\mathcal{E} &\cong \mathcal{O}(4,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 84
$\chi(X, \mathrm{T}_X)$ = -24

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 84
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 20
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 24 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 + x2 + 24 0 x0 + 24 x2
$\mathrm{HH}^{ 3 }$ = x1 + 4 0 0 x1 4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 11 0 0 x1 - x2 - 15 2 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 44t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(667)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(670)

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( 6t + 1 \right) \left( 11t - 1 \right) \left( 4t + 1 \right) ^{2} \left( 5t + 1 \right) ^{2} \left( 496t^{3} + 240t^{2} + 20t - 1 \right) \left( 615875198864288t^{13} + 1300519599193408t^{12} + 1304328936741816t^{11} + 815488003119184t^{10} + 349523964230076t^{9} + 106388812275526t^{8} + 23095255300216t^{7} + 3503805205083t^{6} + 352109391482t^{5} + 20522370029t^{4} + 390334906t^{3} - 21718259t^{2} - 947334t + 2079 \right)D^{6} + 2 \left( 4t + 1 \right) \left( 5t + 1 \right) \left( 4233871007104479713280t^{20} + 12463755326002175410688t^{19} + 17460808283030484976384t^{18} + 15419908459967883406464t^{17} + 9563702629540298820384t^{16} + 4386242735300438050960t^{15} + 1524467414294069396248t^{14} + 404660034537875855812t^{13} + 81563570864685702932t^{12} + 12214944490619403670t^{11} + 1292580872284344710t^{10} + 84696330776998774t^{9} + 1637094380681898t^{8} - 252001562388987t^{7} - 24954528937463t^{6} - 980082803846t^{5} - 18202834085t^{4} - 58163944t^{3} + 30630541t^{2} + 1425159t - 2079 \right)D^{5} + \left( 1411290335701493237760000t^{22} + 4589297304240646652723200t^{21} + 7148268971998874839716608t^{20} + 7073589299252064945632640t^{19} + 4965646848279922131698368t^{18} + 2612661739265637225809824t^{17} + 1060926464696113980535168t^{16} + 337405416538491560213472t^{15} + 84438509815300887270088t^{14} + 16569187667961807990688t^{13} + 2517551315019749157968t^{12} + 288760739493910344302t^{11} + 23793667002476634976t^{10} + 1258411596411577741t^{9} + 27966567188514819t^{8} - 911666873936982t^{7} - 43352725278180t^{6} + 3784344268130t^{5} + 329205578546t^{4} + 7859141440t^{3} - 13619320t^{2} - 1735575t + 2079 \right)D^{4} + 2 t \left( 2963709704973135799296000t^{21} + 9306916387342869742796800t^{20} + 14039618839005223688674560t^{19} + 13483651569253217661112448t^{18} + 9196541999268402582168704t^{17} + 4701053237048102082646848t^{16} + 1852648178858912010072096t^{15} + 570707276848383194952536t^{14} + 137981306215511179527260t^{13} + 26075443717011358217240t^{12} + 3801695067778505360818t^{11} + 416597385904458333938t^{10} + 32590148460188188984t^{9} + 1610068248498872354t^{8} + 28794791918974353t^{7} - 1844634859755623t^{6} - 129302760517420t^{5} - 2198693904940t^{4} + 30715951678t^{3} + 413764668t^{2} - 21603253t - 5049 \right)D^{3} + 4 t ^{2} \left( 3274193578827464311603200t^{20} + 10002484240291448189870080t^{19} + 14712804011007598609883968t^{18} + 13799639604460277058126496t^{17} + 9197407718203425793221424t^{16} + 4592281220313011623580200t^{15} + 1765264897505579418609904t^{14} + 529215402494035527933748t^{13} + 124141371844608418385168t^{12} + 22673041328860106742832t^{11} + 3178535028345487648632t^{10} + 332488993798900396844t^{9} + 24513116460719223804t^{8} + 1102339956534251372t^{7} + 12715483232063727t^{6} - 1682546426045749t^{5} - 101150450623188t^{4} - 2004463903680t^{3} - 4104382971t^{2} - 105315543t - 6207768 \right)D^{2} + 16 t ^{2} \left( 889112911491940739788800t^{20} + 2660583117285962195824640t^{19} + 3839832348926134814560736t^{18} + 3537518922072289765358128t^{17} + 2316470773793857803077352t^{16} + 1135691801605660613116092t^{15} + 428049714932259532219320t^{14} + 125550580094821699710874t^{13} + 28729087320175277243002t^{12} + 5098598670463969489496t^{11} + 690887696438812767097t^{10} + 69293543253308476645t^{9} + 4823224904147507057t^{8} + 195269598375950767t^{7} + 706608650980963t^{6} - 391860285390979t^{5} - 20378841612135t^{4} - 399268399017t^{3} - 2357009142t^{2} - 43556526t - 478170 \right)D + 96 t ^{2} \left( 60483871530063995904000t^{20} + 178436328699396869939200t^{19} + 254160226888236126158272t^{18} + 231239177208689616943136t^{17} + 149547570900161476774416t^{16} + 72367564867465130855304t^{15} + 26888886961474385157936t^{14} + 7760469125277755007836t^{13} + 1742969767823023980932t^{12} + 302590561025113165444t^{11} + 39919865644347145244t^{10} + 3868726646423463530t^{9} + 256217694471960952t^{8} + 9353794185424034t^{7} - 46270873286995t^{6} - 23674358569079t^{5} - 1106947181616t^{4} - 21112779060t^{3} - 162561807t^{2} - 2751705t - 12474 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ6P6

|λ| = 28.89877 λ ≈ 28.89877; multiplicity 1; Jordan blocks 1 λ ≈ 11; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6.072235; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 2, 2 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ -2.826531; multiplicity 1; Jordan blocks 1 2 4 2
Sector dimension
12
Spectral radius
$28.8987662$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28.89877$11
$11$21, 1
$-6.072235$11
$-6$11
$-5$42, 2
$-4$22
$-2.826531$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{4} \left(\lambda + 6\right) \left(\lambda - 11\right)^{2} \left(\lambda^{3} - 20 \lambda^{2} - 240 \lambda - 496\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right)^{2} \left(\lambda + 6\right) \left(\lambda - 11\right) \left(\lambda^{3} - 20 \lambda^{2} - 240 \lambda - 496\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 56 & 12 & 132 & 132 & 492 & 320 & 1184 & 1184 & 3040 \\ 1 & -1 & 0 & 4 & 8 & 48 & 8 & 64 & 120 & 360 & 168 & 864 \\ 1 & 0 & -1 & 4 & 8 & 8 & 48 & 64 & 120 & 168 & 360 & 864 \\ 1 & 1 & 1 & 4 & 0 & 16 & 16 & 72 & 24 & 144 & 144 & 320 \\ 0 & 1 & 1 & 0 & -2 & 4 & 4 & 4 & 40 & 56 & 56 & 240 \\ 0 & 1 & 0 & 1 & 1 & 3 & 0 & 8 & 16 & 56 & 16 & 120 \\ 0 & 0 & 1 & 1 & 1 & 0 & 3 & 8 & 16 & 16 & 56 & 120 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 8 & 8 & 12 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -1 & 2 & 7 & 7 & 40 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 1 & 0 & 1 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 1 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$