Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-320-69-a

·

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

Defining data
Ambient
Fl(1, 2, 7)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w6) + 2 copies of E^(2w2)
Bundle
wedge^4 Q_1^v ot U_2^v ot U_3^v + 2 copies of O(0,2)
ZL62 label
602.577
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,6) \\ \det\mathcal{E} &\cong \mathcal{O}(0,5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 69
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 320
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 320
$\mathrm{c}_1^2\mathrm{c}_2$ = 176
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 0
Hodge diamond
1
00
0 2 0
0 2 2 0
0 0 2 0 0
0 2 2 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 - 4 0 x0 - 4 x2
$\mathrm{HH}^{ 3 }$ = x1 + 92 0 0 x1 92
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 95 0 0 x1 - x2 + 26 0 69
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 64t + 190t^{2} + 64t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(111)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(158)

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( 2t + 1 \right) ^{2} \left( 6896t^{4} + 576t^{3} + 8t^{2} - 16t - 1 \right) \left( 3012961378560t^{9} + 785377115136t^{8} + 98120944896t^{7} + 12849976832t^{6} + 213698784t^{5} - 157592320t^{4} + 32365584t^{3} + 680864t^{2} - 1511t + 180 \right)D^{6} + \left( 2t - 1 \right) \left( 2t + 1 \right) \left( 1745300059990179840t^{15} + 594616202841489408t^{14} - 7811289368088576t^{13} - 22812613330206720t^{12} - 5345246529414144t^{11} - 1387376251498496t^{10} - 147006263386368t^{9} - 3890118977536t^{8} - 3428053103552t^{7} - 321646366464t^{6} - 2343197552t^{5} + 365461248t^{4} - 162933068t^{3} - 2766352t^{2} + 10293t - 360 \right)D^{5} + \left( 58176668666339328000t^{17} + 19657777168825712640t^{16} - 7859406985461399552t^{15} - 3265955720028372992t^{14} - 443260910921760768t^{13} - 51875851068268544t^{12} + 3841934503192576t^{11} + 2305817498498048t^{10} - 295352313144832t^{9} - 84749854598144t^{8} + 6166580756096t^{7} + 408760059968t^{6} + 2753013440t^{5} + 3672780192t^{4} - 136468136t^{3} - 2882324t^{2} - 3227t - 180 \right)D^{4} + t \left( 244342008398625177600t^{16} + 82025546453629009920t^{15} - 14837810891930664960t^{14} - 7425772296018001920t^{13} - 1530680096464846848t^{12} - 300647984803840000t^{11} - 9532292328640512t^{10} + 5278549380096000t^{9} - 649233299726848t^{8} - 135746038109184t^{7} + 17579339604736t^{6} + 1074753207552t^{5} + 16637165760t^{4} + 273648384t^{3} + 11208144t^{2} + 77248t + 9 \right)D^{3} + 8 t ^{2} \left( 67484935652953620480t^{15} + 22541176658811322368t^{14} - 994033167303634944t^{13} - 988768607495673856t^{12} - 277363315992370176t^{11} - 66421819985822720t^{10} - 2386974463313152t^{9} + 919333008905984t^{8} - 170907747768192t^{7} - 22068115129216t^{6} + 2319497903280t^{5} + 112059193640t^{4} + 1468895524t^{3} + 47358180t^{2} + 1721905t + 8120 \right)D^{2} + 16 t ^{2} \left( 36651301259793776640t^{15} + 12197018797928644608t^{14} + 492514047288520704t^{13} - 186112278079107072t^{12} - 90523563562609152t^{11} - 26503247656172544t^{10} - 595314825055488t^{9} + 416172065495552t^{8} - 85311653227008t^{7} - 7870269068800t^{6} + 564393719024t^{5} + 21548327312t^{4} + 258616062t^{3} + 17942292t^{2} + 552351t + 1860 \right)D + 48 t ^{2} \left( 4986571599971942400t^{15} + 1655306224397844480t^{14} + 150527292860841984t^{13} + 2963110029963264t^{12} - 7123272272335872t^{11} - 2717932116495360t^{10} - 5372550195456t^{9} + 53458356147712t^{8} - 10877494026624t^{7} - 795350271872t^{6} + 30574656288t^{5} + 777494512t^{4} + 10748252t^{3} + 1871204t^{2} + 48683t + 120 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 7)

|λ| = 15.93774 λ ≈ -15.93774; multiplicity 1; Jordan blocks 1 λ ≈ 7.937741; multiplicity 1; Jordan blocks 1 λ ≈ -4 - 6.205616i; multiplicity 1; Jordan blocks 1 λ ≈ -4 + 6.205616i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ 2; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
8
Spectral radius
$15.9377413$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-15.93774$11
$7.937741$11
$-4 - 6.205616\mathrm{i}$11
$-4 + 6.205616\mathrm{i}$11
$-2$22
$2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right)^{2} \left(\lambda^{4} + 16 \lambda^{3} - 8 \lambda^{2} - 576 \lambda - 6896\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right)^{2} \left(\lambda^{4} + 16 \lambda^{3} - 8 \lambda^{2} - 576 \lambda - 6896\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 12 & 0 & 0 & 32 & 0 \\ 2 & 0 & 4 & 0 & -40 & 0 & 192 & 32 \\ 1 & 0 & -4 & 2 & 40 & 24 & -192 & -32 \\ 0 & 3 & 2 & 0 & 8 & 0 & -40 & 0 \\ 0 & 0 & 1 & 0 & -8 & 2 & 40 & 12 \\ 0 & 0 & 0 & 3 & 2 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & -4 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$