Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-192-45-h

·

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
E^(w2) + 2 copies of E^(w8) + E^(w1 + w3) + E^(2w3)
Bundle
O(1,0) + 2 copies of wedge^5 Q_1^v ot U_2^v ot wedge^2 U_3^v + U_2^v ot Sigma_(2,1) U_3^v + O(0,2)
ZL62 label
805.77A1w1
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
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 192
$\chi(X, \mathrm{T}_X)$ = -14

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 192
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 32
$\mathrm{c}_4$ = 20
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 14 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 + 40 0 0 x1 - 2 42
$\mathrm{HH}^{ 4 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(389)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(462)

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) ^{2} \left( 3664t^{4} - 576t^{3} - 128t^{2} + t + 1 \right) \left( 20713775472t^{9} + 42468119616t^{8} + 10747478199t^{7} - 1268025318t^{6} - 3206307618t^{5} - 1327655560t^{4} + 76589516t^{3} + 1086680t^{2} - 6420t + 432 \right)D^{6} + \left( t + 1 \right) \left( 986638553282304t^{14} + 2450540818422528t^{13} + 1254496223493072t^{12} + 59654596630656t^{11} - 245570001721152t^{10} - 164223596645451t^{9} - 20173635387554t^{8} + 9095259711679t^{7} - 30905936380t^{6} - 13314766270t^{5} + 7633070728t^{4} - 377812304t^{3} - 4712696t^{2} + 22716t - 864 \right)D^{5} + \left( 5103957131402688t^{15} + 16298595944017920t^{14} + 14690175162181404t^{13} + 4148786442993048t^{12} - 1236123571107096t^{11} - 2093752953640066t^{10} - 869263101370016t^{9} - 60027296725555t^{8} + 22452920351425t^{7} - 1071142591580t^{6} - 74841183258t^{5} - 13439927524t^{4} + 400606484t^{3} + 4601428t^{2} + 23700t + 432 \right)D^{4} + t \left( 13471411015969920t^{14} + 40280156738784768t^{13} + 30195830572866624t^{12} + 6540532187664888t^{11} - 4652208862447824t^{10} - 5791035256802640t^{9} - 2120833817588563t^{8} - 124769890590997t^{7} + 30812375711993t^{6} - 1434860014034t^{5} - 3488923250t^{4} - 487162140t^{3} - 38439472t^{2} - 1030132t - 108 \right)D^{3} + t ^{2} \left( 19182530334007872t^{13} + 54627942005445120t^{12} + 34101891003904716t^{11} + 5515729558841904t^{10} - 8482561915070616t^{9} - 8630377605971488t^{8} - 2802617809365103t^{7} - 105186188920562t^{6} + 26301637548788t^{5} - 202885034120t^{4} + 19996843856t^{3} - 495452224t^{2} - 54454452t - 789216 \right)D^{2} + 24 t ^{2} \left( 583444913719824t^{13} + 1602635476026624t^{12} + 837006759920127t^{11} + 96466478377114t^{10} - 305622701578060t^{9} - 273094144500792t^{8} - 80239246104528t^{7} - 1175099997160t^{6} + 553603479468t^{5} + 30844319520t^{4} + 451765744t^{3} - 14367168t^{2} - 1501824t - 13440 \right)D + 96 t ^{2} \left( 42691091247792t^{13} + 114151947743808t^{12} + 50223526482141t^{11} + 3681361554402t^{10} - 25219866375414t^{9} - 20683551666216t^{8} - 5620031885018t^{7} + 34644026612t^{6} + 32531112860t^{5} + 3867569104t^{4} + 10326392t^{3} - 1137680t^{2} - 97464t - 576 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 9)

|λ| = 11.7777 λ ≈ 11.7777; multiplicity 1; Jordan blocks 1 λ ≈ -8.246097 - 3.969126i; multiplicity 1; Jordan blocks 1 λ ≈ -8.246097 + 3.969126i; multiplicity 1; Jordan blocks 1 λ ≈ 3.714491; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 2; Jordan blocks 2 λ ≈ 0; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
8
Spectral radius
$11.7777019$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$11.7777$11
$-8.246097 - 3.969126\mathrm{i}$11
$-8.246097 + 3.969126\mathrm{i}$11
$3.714491$11
$-1$22
$0$22

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{4} + \lambda^{3} - 128 \lambda^{2} - 576 \lambda + 3664\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{2} \left(\lambda^{4} + \lambda^{3} - 128 \lambda^{2} - 576 \lambda + 3664\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 12 & 24 & 0 & 32 & 80 \\ 1 & -1 & 0 & 8 & 16 & 0 & 0 & 32 \\ 1 & 2 & 0 & -8 & 0 & 24 & 48 & -32 \\ 0 & 1 & -1 & -2 & -2 & 16 & 24 & -12 \\ 0 & 2 & 2 & 1 & 1 & 0 & 8 & 12 \\ 0 & 0 & 0 & 2 & 1 & -2 & -2 & 8 \\ 0 & 0 & 0 & 0 & 2 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 5 & 0\end{pmatrix}\notag\end{equation}$$