Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-54-18-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ6P6
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(2w3) + E^(w1 + 2w3) + E^(w2 + w3)
Bundle
O_2(2) + O_1(1) box O_2(2) + Q_1 box O_2(1)
ZL62 label
20600.Aw7x311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;7) \\ \det\mathcal{E} &\cong \mathcal{O}(2;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 54
$\chi(X, \mathrm{T}_X)$ = -51

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 54
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 168
$\mathrm{c}_1\mathrm{c}_3$ = -20
$\mathrm{c}_4$ = 94
Hodge diamond
1
00
0 2 0
0 0 0 0
0 8 72 8 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 + 51 0 x0 + 51 x2
$\mathrm{HH}^{ 3 }$ = x1 + 3 0 0 x1 3
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 45 0 0 x1 - x2 + 27 0 18
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 26t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(680)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(679)

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( 5t + 1 \right) \left( 4t + 1 \right) ^{4} \left( 41456t^{4} + 10624t^{3} + 255t^{2} + 30t - 1 \right) \left( 60232166588416t^{10} + 83645017751552t^{9} + 50180368395264t^{8} + 17154704797952t^{7} + 3693240667072t^{6} + 518786939904t^{5} + 47630536812t^{4} + 2826609075t^{3} + 98760087t^{2} + 471177t + 729 \right)D^{6} + \left( 4t + 1 \right) ^{3} \left( 1048733573197536952320t^{16} + 2012981226857334571008t^{15} + 1742602044005154553856t^{14} + 902227124134064586752t^{13} + 312045265407911022592t^{12} + 76156059811011438592t^{11} + 13490324015173964032t^{10} + 1757446229652029568t^{9} + 168985466102580576t^{8} + 11902673713080936t^{7} + 594972599167619t^{6} + 19946458265793t^{5} + 479007579852t^{4} + 12300645798t^{3} + 390837663t^{2} + 1417905t + 1458 \right)D^{5} + \left( 4t + 1 \right) ^{2} \left( 34957785773251231744000t^{17} + 70929579966453320253440t^{16} + 65369524912665956188160t^{15} + 36318932264737842266112t^{14} + 13600549907746150268928t^{13} + 3630827677864203825152t^{12} + 711941002664443053056t^{11} + 104076456195592460544t^{10} + 11400617266944236928t^{9} + 930275835621515808t^{8} + 54896587706317372t^{7} + 2153149394933285t^{6} + 45445219410402t^{5} + 178321323897t^{4} - 10638195804t^{3} - 331288677t^{2} - 1039554t - 729 \right)D^{4} + t \left( 4t + 1 \right) \left( 587290800990620693299200t^{17} + 1273637666372298504929280t^{16} + 1262515672270120864972800t^{15} + 759638263854496935837696t^{14} + 310416238577458355568640t^{13} + 91220440567945207775232t^{12} + 19890532037510752460800t^{11} + 3271840886810033502208t^{10} + 408810643522294741248t^{9} + 38726727779045494400t^{8} + 2736073970198852464t^{7} + 137645540094707184t^{6} + 4458965321303493t^{5} + 77394573941559t^{4} + 564711086088t^{3} + 1967284152t^{2} + 5482539t + 4617 \right)D^{3} + 2 t ^{2} \left( 2595266015806171444674560t^{17} + 6057989601580755118456832t^{16} + 6499316031386561611300864t^{15} + 4257340076888987040481280t^{14} + 1906290607948740717969408t^{13} + 618395237215276394872832t^{12} + 150145945116132037378048t^{11} + 27781207766953525026816t^{10} + 3950682284142381371392t^{9} + 432117415606285218688t^{8} + 36014309854932039904t^{7} + 2223738141171981440t^{6} + 95511628745434020t^{5} + 2527516283033547t^{4} + 33670479865263t^{3} + 218635245423t^{2} + 1716535449t + 4026834 \right)D^{2} + 8 t ^{2} \left( 704748961188744831959040t^{17} + 1601458385129760972865536t^{16} + 1670716911685310285348864t^{15} + 1063271247022211097165824t^{14} + 462180807007601068998656t^{13} + 145401113970985369010176t^{12} + 34187649661194974527488t^{11} + 6113500757788882479104t^{10} + 837982870313251676160t^{9} + 88010171725037396800t^{8} + 6995996031197142416t^{7} + 406402608809996712t^{6} + 15979958089550628t^{5} + 367657677836181t^{4} + 3887749577754t^{3} + 22013863668t^{2} + 202005414t + 474579 \right)D + 64 t ^{2} \left( 35956579652486981222400t^{17} + 80204231678425363906560t^{16} + 82051006602282734714880t^{15} + 51163954071918207172608t^{14} + 21773021372740610719744t^{13} + 6699231965872949690368t^{12} + 1538402575953462726656t^{11} + 268167600906393711104t^{10} + 35742051643213642368t^{9} + 3637117818668692576t^{8} + 278373163546563336t^{7} + 15374434887373016t^{6} + 560612159970888t^{5} + 11412598251285t^{4} + 98395639866t^{3} + 477657972t^{2} + 3615354t + 8019 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ6P6

|λ| = 42.44635 λ ≈ 42.44635; multiplicity 1; Jordan blocks 1 λ ≈ -4.18122 - 14.88852i; multiplicity 1; Jordan blocks 1 λ ≈ -4.18122 + 14.88852i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 λ ≈ -4.083906; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
9
Spectral radius
$42.4463463$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$42.44635$11
$-4.18122 - 14.88852\mathrm{i}$11
$-4.18122 + 14.88852\mathrm{i}$11
$-5$11
$-4.083906$11
$-4$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{4} \left(\lambda + 5\right) \left(\lambda^{4} - 30 \lambda^{3} - 255 \lambda^{2} - 10624 \lambda - 41456\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda + 5\right) \left(\lambda^{4} - 30 \lambda^{3} - 255 \lambda^{2} - 10624 \lambda - 41456\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 64 & 792 & 336 & 1272 & 13824 & 6144 & 70080 \\ 1 & -1 & 10 & 44 & 8 & 116 & 1020 & 504 & 6912 \\ 1 & 1 & 4 & 80 & 48 & 124 & 1800 & 864 & 9216 \\ 0 & 1 & 1 & 3 & -2 & 16 & 44 & 8 & 432 \\ 0 & 1 & 0 & 10 & 0 & 12 & 116 & 40 & 504 \\ 0 & 0 & 1 & 1 & 2 & 0 & 80 & 48 & 504 \\ 0 & 0 & 0 & 1 & 0 & \frac{5}{2} & \frac{1}{2} & 1 & 36 \\ 0 & 0 & 0 & 1 & 1 & -\frac{3}{4} & \frac{61}{4} & \frac{5}{2} & 42 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{3} & \frac{2}{3} & 0\end{pmatrix}\notag\end{equation}$$