Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-120-31-a

·

This is a Fano fourfold of K3 type, appearing as case K3-59 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × Fl(1, 2, 3)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)ω1ω2ω3ω4
Bundle
E^(w1 + w2 + w3 + w4)
Bundle
O_1(1) box O_2(1) box O_3(1,1)
ZL62 label
1122.x500
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2,2) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1,1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 31
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 120
$\chi(X, \mathrm{T}_X)$ = -17

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 120
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = 36
Hodge diamond
1
00
0 4 0
0 0 0 0
0 1 24 1 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 17 0 17 x0
$\mathrm{HH}^{ 3 }$ = x0 + 14 0 0 x0 + 7 7
$\mathrm{HH}^{ 4 }$ = 32 0 0 1 0 31
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 26t + 66t^{2} + 26t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(581)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(608)

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( 4t - 1 \right) \left( 5t + 1 \right) \left( 4t + 1 \right) ^{3} \left( 17t^{2} + 2t + 1 \right) \left( 64t^{2} + 16t - 1 \right) \left( 110714428416t^{10} + 183316271508t^{9} + 129059304216t^{8} + 49949272227t^{7} + 11504887043t^{6} + 1603349069t^{5} + 133943337t^{4} + 6807024t^{3} + 207888t^{2} - 748t + 20 \right)D^{6} + \left( 4t + 1 \right) ^{2} \left( 202368260835901440t^{17} + 431975966319599616t^{16} + 408104540983268352t^{15} + 223050828157255296t^{14} + 76965926053254800t^{13} + 16857507960317172t^{12} + 2057978753173848t^{11} + 10818253055433t^{10} - 47957377296117t^{9} - 10365622857300t^{8} - 1368621526702t^{7} - 144764818083t^{6} - 12812927045t^{5} - 791716994t^{4} - 29241180t^{3} - 864780t^{2} + 2284t - 40 \right)D^{5} + \left( 4t + 1 \right) \left( 6745608694530048000t^{18} + 15606385172117422080t^{17} + 16173747545794520064t^{16} + 9869736305717958336t^{15} + 3919539143703057584t^{14} + 1054547121386340592t^{13} + 192710588436435896t^{12} + 22804184787808060t^{11} + 1404279578996629t^{10} - 30387299781148t^{9} - 14452806665205t^{8} - 1133650374564t^{7} + 8933175615t^{6} + 9704049520t^{5} + 890171909t^{4} + 36782348t^{3} + 854688t^{2} - 424t + 20 \right)D^{4} + t \left( 113326226068104806400t^{18} + 284193177807975874560t^{17} + 322310304775330652160t^{16} + 217956704622759618048t^{15} + 97599781310961647360t^{14} + 30398520297511280096t^{13} + 6729884966738783952t^{12} + 1062163380370988324t^{11} + 117458867869052488t^{10} + 8561071658861907t^{9} + 309484930247345t^{8} - 12436383500608t^{7} - 2858775548664t^{6} - 227348374739t^{5} - 9587445177t^{4} - 174048856t^{3} - 3211324t^{2} - 113220t - 52 \right)D^{3} + 2 t ^{2} \left( 125198497370477690880t^{17} + 308619116415903891456t^{16} + 343569024792657770496t^{15} + 227690027442785542272t^{14} + 99780613779675906880t^{13} + 30399448648509997400t^{12} + 6596352652713761992t^{11} + 1028555776684971358t^{10} + 114918768121945270t^{9} + 9010420979369434t^{8} + 455082538482274t^{7} + 8224136379971t^{6} - 831333647609t^{5} - 82845537427t^{4} - 3570490569t^{3} - 99523424t^{2} - 3030674t - 40980 \right)D^{2} + 8 t ^{2} \left( 33997867820431441920t^{17} + 82742069978508607488t^{16} + 90826318248572482560t^{15} + 59257294614557311680t^{14} + 25521175823062369024t^{13} + 7630031206111050744t^{12} + 1623617916713446208t^{11} + 248609568381639782t^{10} + 27423577932749572t^{9} + 2149706393876925t^{8} + 112123127768223t^{7} + 2764058705261t^{6} - 81149887947t^{5} - 8783278552t^{4} - 330702544t^{3} - 13765492t^{2} - 451932t - 3920 \right)D + 16 t ^{2} \left( 6938340371516620800t^{17} + 16739437991888977920t^{16} + 18196917741314832384t^{15} + 11741628034816640640t^{14} + 4993735477465493856t^{13} + 1471969409225234124t^{12} + 308384404561065900t^{11} + 46450526933144727t^{10} + 5039562607071405t^{9} + 388133572963720t^{8} + 19712769861152t^{7} + 454306361441t^{6} - 13172690381t^{5} - 1099167280t^{4} - 33446856t^{3} - 1837716t^{2} - 51516t - 320 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × Fl(1, 2, 3)

|λ| = 19.31371 λ ≈ 19.31371; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ -1 - 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1 + 4i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 5; Jordan blocks 1, 1, 1, 2 λ ≈ 4; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3.313708; multiplicity 1; Jordan blocks 1 4 2 2 5 2
Sector dimension
17
Spectral radius
$19.3137085$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$19.31371$11
$-5$41, 1, 1, 1
$-1 - 4\mathrm{i}$21, 1
$-1 + 4\mathrm{i}$21, 1
$-4$51, 1, 1, 2
$4$21, 1
$-3.313708$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{5} \left(\lambda + 5\right)^{4} \left(\lambda - 4\right)^{2} \left(\lambda^{2} + 2 \lambda + 17\right)^{2} \left(\lambda^{2} - 16 \lambda - 64\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 5\right) \left(\lambda - 4\right) \left(\lambda^{2} + 2 \lambda + 17\right) \left(\lambda^{2} - 16 \lambda - 64\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 6 & 6 & 10 & 10 & 12 & 24 & 24 & 24 & 24 & 24 & 24 & 72 & 72 & 64 & 64 & 160 \\ 1 & -1 & 1 & 1 & 1 & 4 & 8 & 8 & 2 & 2 & 4 & 4 & 18 & 18 & 18 & 12 & 40 \\ 1 & 1 & -1 & 1 & 1 & 4 & 2 & 2 & 8 & 8 & 4 & 4 & 18 & 18 & 12 & 18 & 40 \\ 1 & 1 & 1 & -1 & 2 & 2 & 4 & 4 & 4 & 4 & 6 & 2 & 6 & 12 & 12 & 12 & 24 \\ 1 & 1 & 1 & 2 & -1 & 2 & 4 & 4 & 4 & 4 & 2 & 6 & 12 & 6 & 6 & 6 & 24 \\ 0 & 1 & 1 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 0 & 0 & 6 & 6 & 4 & 4 & 12 \\ 0 & 1 & 0 & 1 & 0 & 1 & -2 & 2 & 1 & 0 & 1 & 0 & 2 & 4 & 4 & 2 & 6 \\ 0 & 1 & 0 & 0 & 1 & 1 & 2 & -2 & 0 & 1 & 0 & 1 & 4 & 2 & 2 & 0 & 6 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & -2 & 2 & 1 & 0 & 2 & 4 & 2 & 4 & 6 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 2 & -2 & 0 & 1 & 4 & 2 & 0 & 2 & 6 \\ 0 & 0 & 0 & 1 & 2 & 0 & 1 & 2 & 1 & 2 & -1 & 1 & 2 & 4 & 4 & 4 & 6 \\ 0 & 0 & 0 & 2 & 1 & 0 & 2 & 1 & 2 & 1 & 1 & -1 & 4 & 2 & 2 & 2 & 6 \\ 0 & 0 & 0 & 0 & 0 & 1 & 3 & 1 & 3 & 1 & \frac{2}{3} & \frac{8}{3} & 1 & 4 & \frac{8}{3} & \frac{8}{3} & \frac{22}{3} \\ 0 & 0 & 0 & 0 & 0 & 1 & -2 & 0 & -2 & 0 & -\frac{4}{3} & -\frac{10}{3} & -2 & -5 & -\frac{7}{3} & -\frac{7}{3} & -\frac{14}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 0 & 0 & 2 & 2 & 3 & 3 & 0 & 2 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 2 & 2 & 3 & 3 & 2 & 0 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$