Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-256-57-c

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × Fl(1, 3, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7 (marked)ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w4 + w5) + E^(w1 + w3) + E^(w2 + w7)
Bundle
Q_2 box O_3(1,0) + O_1(1) box O_2(1) + Q_1 box O_3(0,1)
ZL62 label
202034.wHw8w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 57
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 256
$\chi(X, \mathrm{T}_X)$ = 2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 256
$\mathrm{c}_1^2\mathrm{c}_2$ = 160
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 2 0 x0 - 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 38 0 0 x1 - 16 54
$\mathrm{HH}^{ 4 }$ = 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(281)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(360)

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( 10t - 1 \right) \left( 2t + 1 \right) ^{2} \left( 695t^{5} + 33t^{4} + 83t^{3} + 6t^{2} + 5t + 1 \right) \left( 931521421231200t^{12} + 1617974810665492t^{11} + 1845516915183602t^{10} + 1322681700656773t^{9} + 530037841839958t^{8} + 113917113681548t^{7} + 11375218530063t^{6} + 523583209685t^{5} + 45666268681t^{4} + 4973082137t^{3} + 83199814t^{2} - 6829287t + 43136 \right)D^{6} + \left( 2t + 1 \right) \left( 1631466617144323680000t^{20} + 3663239815135876689600t^{19} + 4980688157039727901920t^{18} + 4533184379037930327472t^{17} + 2719405709297102281468t^{16} + 1119166385816756279328t^{15} + 338924429653542773881t^{14} + 81649715553565253227t^{13} + 16102393941686955439t^{12} + 2510804662015085748t^{11} + 325988920539018428t^{10} + 46258441611019310t^{9} + 8558784025418586t^{8} + 1330422441615091t^{7} + 115204892310405t^{6} + 4898737781100t^{5} + 256350605128t^{4} + 22135595627t^{3} + 412638775t^{2} - 21005493t + 86272 \right)D^{5} + \left( 27191110285738728000000t^{21} + 72350294203555717584000t^{20} + 109383412563114191570720t^{19} + 111758251136700151165568t^{18} + 77116963315027147417840t^{17} + 36285178524839746361656t^{16} + 11977688873212040575198t^{15} + 2895109591311806288945t^{14} + 538384999123996128317t^{13} + 77383481367769785911t^{12} + 7702092692069635821t^{11} + 349033255805300349t^{10} - 36045158898307784t^{9} - 11528122613363697t^{8} - 1748682249307051t^{7} - 142424106016146t^{6} - 6225693827137t^{5} - 421486387223t^{4} - 32067009529t^{3} - 629983733t^{2} + 11088949t - 43136 \right)D^{4} + t \left( 114202663200102657600000t^{20} + 296283684722725762752000t^{19} + 446778464384053825401600t^{18} + 453962898931631077061920t^{17} + 306442078525595464859640t^{16} + 138164616717089175687068t^{15} + 42448190165562833255806t^{14} + 9170509634736946136141t^{13} + 1466807170895924304367t^{12} + 180922803084368362901t^{11} + 16775500837233876200t^{10} + 1225328930060949836t^{9} + 125371180557168892t^{8} + 17224577778616070t^{7} + 1560201536711797t^{6} + 60886698482311t^{5} - 465167409858t^{4} - 86522241064t^{3} - 1874696949t^{2} + 90510183t + 24601 \right)D^{3} + 2 t ^{2} \left( 126166751725827697920000t^{19} + 320909640786692439943200t^{18} + 483576375452603379415360t^{17} + 489600617813270638319832t^{16} + 324789633028308562109288t^{15} + 141597955224461609471894t^{14} + 41149209320279192782118t^{13} + 8157778958160770903684t^{12} + 1165604775788997883118t^{11} + 131234192736741852417t^{10} + 12788224444632002723t^{9} + 1264470098300299144t^{8} + 150628902811803663t^{7} + 15322480061154143t^{6} + 927501260714568t^{5} + 19267395202900t^{4} - 818307822445t^{3} - 19483559079t^{2} + 516487539t + 31595264 \right)D^{2} + 4 t ^{2} \left( 68521597920061594560000t^{19} + 171734658955742530639200t^{18} + 258833770038906149715840t^{17} + 261437017150469115330024t^{16} + 171115788857104031201328t^{15} + 72694145219223993295758t^{14} + 20245119759176477788320t^{13} + 3762329719908089686885t^{12} + 496448721181260549299t^{11} + 53732792403998324689t^{10} + 5781104866060686872t^{9} + 680276948522801084t^{8} + 79571159550211946t^{7} + 6679636922728111t^{6} + 301128419265324t^{5} + 2859645220851t^{4} - 284078825429t^{3} + 1620089786t^{2} + 407591148t + 6039040 \right)D + 48 t ^{2} \left( 2330666595920462400000t^{19} + 5782660393267307388000t^{18} + 8719175871014796053760t^{17} + 8793431951923708960584t^{16} + 5701126222553255144640t^{15} + 2377825411201072342158t^{14} + 642360963032991389838t^{13} + 113993293990015461100t^{12} + 14263066688353832834t^{11} + 1529987577224155483t^{10} + 179289305860960655t^{9} + 22694642182840832t^{8} + 2534312137542677t^{7} + 188519302150291t^{6} + 6954874799562t^{5} + 23576339904t^{4} - 4947783467t^{3} + 189425453t^{2} + 11714457t + 86272 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × Fl(1, 3, 4)

|λ| = 10 λ ≈ 10; multiplicity 1; Jordan blocks 1 λ ≈ -6.386525; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ 1.622778 - 2.884794i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.622778 + 2.884794i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.9295156 - 3.011515i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.9295156 + 3.011515i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 1, 1, 1 2 2 2 2 2 4
Sector dimension
16
Spectral radius
$10$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10$11
$-6.386525$21, 1
$-6$11
$1.622778 - 2.884794\mathrm{i}$21, 1
$1.622778 + 2.884794\mathrm{i}$21, 1
$-0.9295156 - 3.011515\mathrm{i}$21, 1
$-0.9295156 + 3.011515\mathrm{i}$21, 1
$-2$41, 1, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda + 6\right) \left(\lambda - 10\right) \left(\lambda^{5} + 5 \lambda^{4} + 6 \lambda^{3} + 83 \lambda^{2} + 33 \lambda + 695\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 6\right) \left(\lambda - 10\right) \left(\lambda^{5} + 5 \lambda^{4} + 6 \lambda^{3} + 83 \lambda^{2} + 33 \lambda + 695\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 4 & 4 & 0 & 3 & 3 & 0 & 3 & 3 & 12 & 12 & 8 & 12 & 20 \\ 1 & -1 & 0 & 0 & 1 & 2 & 4 & 2 & 0 & 2 & 0 & 0 & 3 & -3 & 3 & 0 \\ 1 & 0 & -1 & 1 & 0 & 2 & 0 & 2 & 2 & 2 & 4 & 3 & 0 & -3 & 6 & 0 \\ 1 & 0 & 1 & -1 & 1 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 3 & 3 & 0 & 8 \\ 1 & 1 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 2 & 2 & 3 & 0 & 3 & 0 & 8 \\ 0 & 1 & 2 & -1 & -1 & -2 & 0 & 0 & -1 & 1 & 0 & 2 & 4 & 2 & 2 & 6 \\ 0 & 1 & 0 & 2 & 1 & 1 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 2 & 0 & 1 & -1 & 0 & 0 & 0 & 2 & 2 & 0 & 2 & 0 \\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 2 & -2 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 2 & 2 & 0 & 4 & 0 \\ 0 & 0 & 1 & 1 & 2 & 1 & 1 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & -1 & -\frac{1}{2} & \frac{9}{2} & 2 & -\frac{1}{2} & 1 & \frac{1}{2} & \frac{1}{2} & 2 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & -\frac{3}{2} & \frac{7}{2} & 5 & \frac{5}{2} & -2 & \frac{1}{2} & -\frac{1}{2} & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 1 & 1 & \frac{5}{2} & -\frac{7}{2} & -2 & -\frac{1}{2} & 1 & -\frac{3}{2} & \frac{3}{2} & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 2 & -3 & -2 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{5}{2} & 1 & 3 & 0\end{pmatrix}\notag\end{equation}$$