Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-193-45-b

·

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

Defining data
Ambient
Fl(2, 3, 9)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 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
2 copies of E^(w8) + 2 copies of E^(w1 + w3)
Bundle
2 copies of wedge^5 Q_1^v ot U_2^v ot wedge^2 U_3^v + 2 copies of U_2^v ot Sigma_(2,1) U_3^v
ZL62 label
805.77w1w1
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 193
$\chi(X, \mathrm{T}_X)$ = -14

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 193
$\mathrm{c}_1^2\mathrm{c}_2$ = 142
$\mathrm{c}_2^2$ = 112
$\mathrm{c}_1\mathrm{c}_3$ = 30
$\mathrm{c}_4$ = 21
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 15 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 + 14 0 x0 + 14 x1
$\mathrm{HH}^{ 3 }$ = x1 + 39 0 0 x1 - 5 44
$\mathrm{HH}^{ 4 }$ = 45 0 0 0 0 45
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 111t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(374)
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( t - 1 \right) \left( 15760t^{6} - 20944t^{5} + 4800t^{4} - 88t^{3} - 132t^{2} - t + 1 \right) \left( 89208307850000t^{12} - 382667805726400t^{11} + 824487130855180t^{10} - 455312717510868t^{9} - 22437481658288t^{8} + 99837172270667t^{7} - 39685799024756t^{6} + 6159542219166t^{5} - 159741058982t^{4} - 80910243742t^{3} + 56279284t^{2} - 104279649t - 654656 \right)D^{6} + 2 \left( 9138499056154000000t^{19} - 56950270118705248000t^{18} + 173423021267217204000t^{17} - 253618918580013859200t^{16} + 173431019279453015872t^{15} - 33475212295451384360t^{14} - 29573377202764562792t^{13} + 24755153654830996430t^{12} - 8808590103262256380t^{11} + 1675200031403699698t^{10} - 133458253746036785t^{9} - 9771403210998212t^{8} + 2741644969311295t^{7} - 225744467034195t^{6} + 17424542215000t^{5} - 347061390991t^{4} - 269978656660t^{3} - 73701630t^{2} - 209213954t - 981984 \right)D^{5} + \left( 94548317157901000000t^{19} - 563720346470252480000t^{18} + 1675639285352452367600t^{17} - 2190531617312603704560t^{16} + 1200549990433588823760t^{15} + 2534716498624612852t^{14} - 378738248533476857196t^{13} + 237698353760597249228t^{12} - 74283376070726611450t^{11} + 12051567842008022654t^{10} - 558958521673663539t^{9} - 114263225755193102t^{8} + 8304830196438179t^{7} + 852340347736674t^{6} - 61524637349052t^{5} + 1361316366006t^{4} + 603707154427t^{3} + 3541430124t^{2} + 467221987t + 1309312 \right)D^{4} + 2 t \left( 124775660189795000000t^{18} - 721043519203823680000t^{17} + 2109820738297834368000t^{16} - 2477882943627773040400t^{15} + 1059824749933952066640t^{14} + 277819077981769999404t^{13} - 541543036317656365656t^{12} + 289740870515044015714t^{11} - 81631550816312623649t^{10} + 11098114592164769663t^{9} + 37496531322699820t^{8} - 212174618132329256t^{7} + 14692858556020886t^{6} - 192893227927751t^{5} + 52907060282182t^{4} - 916369542173t^{3} - 36743632637t^{2} - 370480123t + 8480 \right)D^{3} + 4 t ^{2} \left( 88836755247804750000t^{17} - 502428284708650784000t^{16} + 1456213430279440505900t^{15} - 1555287503889647103700t^{14} + 503567749463147712412t^{13} + 316983074114855129159t^{12} - 397564348932914279899t^{11} + 191321255054769858164t^{10} - 49062418541288384257t^{9} + 5399737482108836567t^{8} + 392045255490363429t^{7} - 168801832276378607t^{6} + 9987365508985774t^{5} - 150333581820223t^{4} + 24384896434405t^{3} - 698183400516t^{2} - 12407065808t - 98618368 \right)D^{2} + 8 t ^{2} \left( 32424097612700250000t^{17} - 180705352170941368000t^{16} + 520949586745345324500t^{15} - 513557694151572475100t^{14} + 121021582052038022372t^{13} + 143633666106368752509t^{12} - 147098120203727665021t^{11} + 65596395629120278346t^{10} - 15421805336484655658t^{9} + 1318203174137761278t^{8} + 225136535226390242t^{7} - 62295174592738166t^{6} + 3408453077618828t^{5} - 65004967060350t^{4} + 4268938330159t^{3} - 243923299987t^{2} - 3123214104t - 10474496 \right)D + 96 t ^{3} \left( 790831649090250000t^{16} - 4363455878730820000t^{15} + 12544325746059390900t^{14} - 11583942609153025340t^{13} + 1879182209494211180t^{12} + 3964918402108363917t^{11} - 3617714311455974461t^{10} + 1525086713510334726t^{9} - 331330887825621614t^{8} + 20669965598411582t^{7} + 6755720074892190t^{6} - 1471434117271109t^{5} + 76480677732165t^{4} - 1807367891472t^{3} + 21636906216t^{2} - 5486137308t - 69598116 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 9)

|λ| = 11.26625 λ ≈ 11.26625; multiplicity 1; Jordan blocks 1 λ ≈ -8.781489 - 3.492064i; multiplicity 1; Jordan blocks 1 λ ≈ -8.781489 + 3.492064i; multiplicity 1; Jordan blocks 1 λ ≈ 3.172874 - 2.530528i; multiplicity 1; Jordan blocks 1 λ ≈ 3.172874 + 2.530528i; multiplicity 1; Jordan blocks 1 λ ≈ 1; multiplicity 1; Jordan blocks 1 λ ≈ 0.950975; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 2 2
Sector dimension
9
Spectral radius
$11.2662531$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.26625$11
$-8.781489 - 3.492064\mathrm{i}$11
$-8.781489 + 3.492064\mathrm{i}$11
$3.172874 - 2.530528\mathrm{i}$11
$3.172874 + 2.530528\mathrm{i}$11
$1$11
$0.950975$11
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda - 1\right) \left(\lambda^{6} - \lambda^{5} - 132 \lambda^{4} - 88 \lambda^{3} + 4800 \lambda^{2} - 20944 \lambda + 15760\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda - 1\right) \left(\lambda^{6} - \lambda^{5} - 132 \lambda^{4} - 88 \lambda^{3} + 4800 \lambda^{2} - 20944 \lambda + 15760\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 12 & 12 & 6 & 0 & 32 & 80 \\ 1 & 0 & 0 & 8 & 16 & 8 & -24 & -36 & 32 \\ 1 & 0 & 0 & -8 & 0 & 8 & 60 & 84 & -32 \\ 0 & 1 & 0 & 1 & 0 & 1 & 16 & 24 & 0 \\ 0 & 2 & 1 & -3 & 0 & -1 & 0 & 8 & -12 \\ 0 & 0 & 1 & 4 & 0 & 1 & 0 & 0 & 24 \\ 0 & 0 & 0 & 2 & 1 & -\frac{1}{4} & 0 & 0 & 8 \\ 0 & 0 & 0 & 0 & 2 & \frac{7}{4} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 5 & 0\end{pmatrix}\notag\end{equation}$$