Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-146-36-b

·

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

Defining data
Ambient
Gr(2, 4) × Gr(2, 6)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
3 copies of E^(w5) + E^(w1 + w4) + E^(w2)
Bundle
3 copies of O_2(1) + U_1^v box U_2^v + O_1(1)
ZL62 label
3151.4441w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(3;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 146
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 146
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 110
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 0 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 + x1 + 11 0 x0 + 11 x1
$\mathrm{HH}^{ 3 }$ = x1 + 32 0 0 x1 + 13 19
$\mathrm{HH}^{ 4 }$ = 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(483)
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( 2t + 1 \right) ^{2} \left( 31t^{4} + 6t^{3} - 7t^{2} - 6t - 1 \right) \left( 1823t^{4} + 1086t^{3} + 129t^{2} + 2t - 1 \right) \left( 58070084218357048t^{17} + 180296104095870144t^{16} + 231841341116628020t^{15} + 137627345980784838t^{14} + 587209953998761t^{13} - 63515014289648531t^{12} - 53708763649336172t^{11} - 24874307888641764t^{10} - 7377866083877236t^{9} - 1422414174064228t^{8} - 166545213614430t^{7} - 8752155986750t^{6} + 322339636336t^{5} + 63996034532t^{4} + 927180028t^{3} - 228708458t^{2} - 5123703t - 9271 \right)D^{6} + \left( 2t + 1 \right) \left( 137832016116144497852208t^{26} + 579046908096662493949208t^{25} + 1054771067980466810059224t^{24} + 1012745308182687786128444t^{23} + 397107437965050128600178t^{22} - 240182637085036696349995t^{21} - 453398566118993187802278t^{20} - 309792404407327398897430t^{19} - 103741866288102791704550t^{18} + 4087545216593336182635t^{17} + 24547691897265227688414t^{16} + 14746181666423971830128t^{15} + 5306460545356272849634t^{14} + 1318696391135541242205t^{13} + 232056634102656376430t^{12} + 28248197510919851906t^{11} + 2212134786499721234t^{10} + 98396111726929977t^{9} + 4132192756710522t^{8} + 722786438267884t^{7} + 80499626696522t^{6} + 2293232990395t^{5} - 167797935434t^{4} - 3192513818t^{3} + 849814194t^{2} + 15185689t + 18542 \right)D^{5} + \left( 2297200268602408297536800t^{27} + 10444757480102631606786080t^{26} + 20949852228109009023604840t^{25} + 23149858839334016623994632t^{24} + 12909998930404043564178040t^{23} - 1165112288382751195065206t^{22} - 8706491197601813264303505t^{21} - 8107364694224524003710715t^{20} - 4248844513521965145625644t^{19} - 1289395472048385480307008t^{18} - 95624467169232277575363t^{17} + 117662337392534987262467t^{16} + 69191123711244702457706t^{15} + 21680759323841342772518t^{14} + 4462455282651015638685t^{13} + 600794043395784649773t^{12} + 42796238575782463932t^{11} - 1213871858483786504t^{10} - 676974482031826283t^{9} - 76424418381378527t^{8} - 3706293734436584t^{7} + 45802752611762t^{6} + 15795682118919t^{5} + 805787977131t^{4} + 13159406858t^{3} - 757311452t^{2} - 11298585t - 9271 \right)D^{4} + t \left( 9648241128130114849654560t^{26} + 42697720712259326192753440t^{25} + 83198731445637722848427960t^{24} + 88660386249617991136278384t^{23} + 45855215214644581418523104t^{22} - 9129368465428955333302068t^{21} - 36945568557828531169421653t^{20} - 33499739361794881314482780t^{19} - 18202597212903726422333678t^{18} - 6490656801547190745660256t^{17} - 1390147177431202832693011t^{16} - 61611735165961567876600t^{15} + 73214209870989010150496t^{14} + 29618380177830297469796t^{13} + 6248190610415024194819t^{12} + 795285291468430312500t^{11} + 51715374624588340422t^{10} - 438423649915271208t^{9} - 238573835906227317t^{8} + 25190754656616160t^{7} + 8528387116675580t^{6} + 854070164534388t^{5} + 40468258448529t^{4} + 892037974604t^{3} + 9926794350t^{2} + 16868016t + 12827 \right)D^{3} + 2 t ^{2} \left( 10659009246315174500570752t^{25} + 46181666334541343871778624t^{24} + 87872394745977087471753704t^{23} + 90601434459047458249877288t^{22} + 43093525819789288540733164t^{21} - 14600298608765170617989336t^{20} - 42073893280830521144046846t^{19} - 36962389364048321753847548t^{18} - 20226443879739285741148741t^{17} - 7612973369684149170281498t^{16} - 1960655057626323904289568t^{15} - 304961064577138766048372t^{14} - 8329122362085299913899t^{13} + 9039171040500577815242t^{12} + 2509807500311245507941t^{11} + 350815299627445765312t^{10} + 28325560845473917725t^{9} + 1710206821135715844t^{8} + 281888903863421903t^{7} + 61222317537952820t^{6} + 7811658128976410t^{5} + 555929988238306t^{4} + 20583657423945t^{3} + 370539748312t^{2} + 3682382940t + 10779326 \right)D^{2} + 4 t ^{2} \left( 5788944676878068909792736t^{25} + 24687743230855358682670624t^{24} + 46116825219335142278580416t^{23} + 46262748361181148752536088t^{22} + 20305434825368937219954932t^{21} - 9876153185961275757314736t^{20} - 23470411428102798205004849t^{19} - 20060308880349834847476826t^{18} - 10933241315598207457231961t^{17} - 4186109168432324253728270t^{16} - 1142908475604567126268999t^{15} - 213766979345167542610190t^{14} - 22951551970830124087974t^{13} + 146495985898395785608t^{12} + 536435373946090610865t^{11} + 104160408716661029248t^{10} + 12844121660783243919t^{9} + 1577998723172413572t^{8} + 239420057096314506t^{7} + 33105096391285742t^{6} + 3186549950664862t^{5} + 186052760399918t^{4} + 5777708202386t^{3} + 91524158746t^{2} + 706902539t + 2558796 \right)D + 16 t ^{2} \left( 590708640497762133652320t^{25} + 2492017845816137561410080t^{24} + 4595095424895027688994640t^{23} + 4517208569520705915235752t^{22} + 1855713856715370527186620t^{21} - 1146054880199716162027584t^{20} - 2439973264596252773669525t^{19} - 2044187089447642548523346t^{18} - 1107235234710218680231591t^{17} - 425711001317808950763378t^{16} - 118681536838619000123648t^{15} - 23553726777248388143900t^{14} - 3061389209869446317941t^{13} - 171882890377961267642t^{12} + 23694730244983828877t^{11} + 7768622525869831640t^{10} + 1292802403499307421t^{9} + 188450034684863224t^{8} + 26275593179378400t^{7} + 3054512206336910t^{6} + 251983408277024t^{5} + 12800912101810t^{4} + 347516224837t^{3} + 5108651188t^{2} + 27645360t + 92710 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 6)

|λ| = 15.42461 λ ≈ 15.42461; multiplicity 1; Jordan blocks 1 λ ≈ -5.600877 - 4.669004i; multiplicity 1; Jordan blocks 1 λ ≈ -5.600877 + 4.669004i; multiplicity 1; Jordan blocks 1 λ ≈ -4.484435; multiplicity 1; Jordan blocks 1 λ ≈ -2.222852; multiplicity 1; Jordan blocks 1 λ ≈ -1.5 - 1.551404i; multiplicity 1; Jordan blocks 1 λ ≈ -1.5 + 1.551404i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ 1.484435; multiplicity 1; Jordan blocks 1 2
Sector dimension
10
Spectral radius
$15.4246064$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$15.42461$11
$-5.600877 - 4.669004\mathrm{i}$11
$-5.600877 + 4.669004\mathrm{i}$11
$-4.484435$11
$-2.222852$11
$-1.5 - 1.551404\mathrm{i}$11
$-1.5 + 1.551404\mathrm{i}$11
$-2$22
$1.484435$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{4} + 6 \lambda^{3} + 7 \lambda^{2} - 6 \lambda - 31\right) \left(\lambda^{4} - 2 \lambda^{3} - 129 \lambda^{2} - 1086 \lambda - 1823\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{4} + 6 \lambda^{3} + 7 \lambda^{2} - 6 \lambda - 31\right) \left(\lambda^{4} - 2 \lambda^{3} - 129 \lambda^{2} - 1086 \lambda - 1823\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 12 & 12 & 60 & 30 & 54 & 72 & 16 & 100 \\ 1 & -1 & 3 & 2 & 6 & 4 & 6 & 30 & 12 & 32 \\ 1 & 1 & 0 & 4 & 16 & 6 & 12 & 12 & 0 & 16 \\ 0 & 2 & 0 & -1 & 6 & 1 & 3 & -2 & -4 & 0 \\ 0 & 1 & 1 & 0 & -1 & 2 & 3 & 6 & 4 & 12 \\ 0 & 0 & 1 & 1 & 1 & -2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & -2 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 7 & \frac{9}{2} & \frac{15}{2} & \frac{3}{2} & 1 & 8 \\ 0 & 0 & 0 & 1 & -\frac{11}{2} & -\frac{25}{4} & -\frac{39}{4} & -\frac{3}{4} & -\frac{5}{2} & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$