Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-350-75-a

·

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

Defining data
Ambient
ℙ3P3 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3 + w5)
Bundle
Q_1 box O_2(1)
ZL62 label
3031.q
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 75
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 350
$\chi(X, \mathrm{T}_X)$ = 7

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 350
$\mathrm{c}_1^2\mathrm{c}_2$ = 188
$\mathrm{c}_2^2$ = 90
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 7 0 x0 - 7 x1
$\mathrm{HH}^{ 3 }$ = x1 + 57 0 0 x1 - 38 95
$\mathrm{HH}^{ 4 }$ = 75 0 0 0 0 75
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 70t + 208t^{2} + 70t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(43)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(67)

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) \left( 5t + 1 \right) \left( 7t^{2} + t + 1 \right) \left( 28t^{2} + 4t - 1 \right) \left( 36t^{2} - 4t + 1 \right) \left( 287259243264t^{9} + 263604336128t^{8} + 190465972608t^{7} + 66420142144t^{6} + 9746703392t^{5} + 796858224t^{4} + 47663800t^{3} + 2086700t^{2} - 13401t + 175 \right)D^{6} + \left( 425649256298864640t^{17} + 710563104203452416t^{16} + 693037963829772288t^{15} + 446473070182236160t^{14} + 191472771723184640t^{13} + 58450506402152704t^{12} + 12998040487544320t^{11} + 1987692375493632t^{10} + 202694660110240t^{9} + 16706495054160t^{8} + 2572249361856t^{7} + 573423049728t^{6} + 69899291174t^{5} + 4920625087t^{4} + 237199670t^{3} + 8363600t^{2} - 40203t + 350 \right)D^{5} + \left( 3547077135823872000t^{17} + 5572629550201835520t^{16} + 5316234437264859136t^{15} + 3272336361974077440t^{14} + 1294000694655443456t^{13} + 352150685176035584t^{12} + 69627825378108672t^{11} + 9830019881114880t^{10} + 963971222538592t^{9} + 58598211879216t^{8} - 1020260577216t^{7} - 984488961256t^{6} - 115351465578t^{5} - 6490085261t^{4} - 230086559t^{3} - 7471716t^{2} + 36528t - 175 \right)D^{4} + t \left( 14897723970460262400t^{16} + 22254236260918824960t^{15} + 20965461097611018240t^{14} + 12461656835922051072t^{13} + 4595836379950034432t^{12} + 1131824636048419584t^{11} + 201209599906602496t^{10} + 26320831572519424t^{9} + 2594888959691232t^{8} + 189916515993840t^{7} + 9726777858432t^{6} + 258998965632t^{5} - 5040915966t^{4} + 50101861t^{3} - 7029762t^{2} + 70238t + 74 \right)D^{3} + 2 t ^{2} \left( 16458437910222766080t^{15} + 23612925805575192576t^{14} + 22104409918586089472t^{13} + 12784907229302921216t^{12} + 4446425214127825920t^{11} + 1009389505909071360t^{10} + 164396204957229056t^{9} + 20129710898197248t^{8} + 1939939458840192t^{7} + 144373226046144t^{6} + 7205800021008t^{5} + 172127445928t^{4} - 4809883124t^{3} + 14587530t^{2} - 1227504t + 26425 \right)D^{2} + 4 t ^{2} \left( 8938634382276157440t^{15} + 12437126421380591616t^{14} + 11610169173947731968t^{13} + 6576885772579985408t^{12} + 2182144574079919616t^{11} + 465227579002286848t^{10} + 70869994890647040t^{9} + 8239712741423616t^{8} + 773102776308768t^{7} + 57372128472720t^{6} + 2774361380784t^{5} + 60553733064t^{4} - 2038587590t^{3} - 4305083t^{2} - 217270t + 6300 \right)D + 16 t ^{2} \left( 912105549211852800t^{15} + 1242406039057182720t^{14} + 1158587845465042944t^{13} + 646772400561702912t^{12} + 207324167961670144t^{11} + 42261166612305152t^{10} + 6140170675086592t^{9} + 687441103315584t^{8} + 62811138168672t^{7} + 4593599232624t^{6} + 216533143200t^{5} + 4600663920t^{4} - 148688478t^{3} - 582631t^{2} - 19526t + 350 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 4)

|λ| = 7.656854 λ ≈ 7.656854; multiplicity 1; Jordan blocks 1 λ ≈ 2 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ 2 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 λ ≈ -3.656854; multiplicity 1; Jordan blocks 1 λ ≈ -0.5 - 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -0.5 + 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
9
Spectral radius
$7.65685425$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.656854$11
$2 - 5.656854\mathrm{i}$11
$2 + 5.656854\mathrm{i}$11
$-5$11
$-3.656854$11
$-0.5 - 2.598076\mathrm{i}$11
$-0.5 + 2.598076\mathrm{i}$11
$-2$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 5\right) \left(\lambda^{2} + \lambda + 7\right) \left(\lambda^{2} - 4 \lambda + 36\right) \left(\lambda^{2} - 4 \lambda - 28\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 5\right) \left(\lambda^{2} + \lambda + 7\right) \left(\lambda^{2} - 4 \lambda + 36\right) \left(\lambda^{2} - 4 \lambda - 28\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 0 & 0 & 0 & 4 & 8 & 10 \\ 3 & 0 & 2 & 4 & 0 & 2 & 0 & -3 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 \\ 0 & 1 & 4 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 2 & 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 8 & 2 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & 0\end{pmatrix}\notag\end{equation}$$