Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 6-230-51-a

·

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

Defining data
Ambient
Fl(2, 3, 5)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1 + w4)
Bundle
Q_1^v ot U_2^v ot Sigma_(2,1) U_3^v
ZL62 label
45.U
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 230
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 230
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 30
Hodge diamond
1
00
0 6 0
0 0 0 0
0 0 16 0 0
0 0 0 0
0 6 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 4 0 x0 - 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 18 0 0 x1 - 26 44
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 46t + 136t^{2} + 46t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
Coates et al. (2020) Obro(4,40)
Coates–Kasprzyk–Prince (2015) CKP(499)

Quantum differential equation

With $D=t\frac{\mathrm{d}}{\mathrm{d}t}$, the regularized period satisfies $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 2t - 1 \right) \left( 3t - 1 \right) \left( 6t + 1 \right) \left( 10t - 1 \right) \left( 90t - 23 \right) ^{2} \left( 25t^{2} - 5t - 1 \right)D^{4} + 2 t \left( 90t - 23 \right) \left( 4050000t^{6} - 4374000t^{5} + 1505400t^{4} - 112460t^{3} - 32199t^{2} + 4784t + 2 \right)D^{3} + t \left( 2551500000t^{7} - 3148470000t^{6} + 1436058000t^{5} - 271852200t^{4} + 7955470t^{3} + 3791849t^{2} - 359900t - 46 \right)D^{2} + 6 \left( 90t - 23 \right) t ^{2} \left( 6750000t^{5} - 6138000t^{4} + 1842200t^{3} - 157860t^{2} - 14333t + 2024 \right)D + 80 t ^{2} \left( 21870000t^{6} - 24543000t^{5} + 10275300t^{4} - 1843050t^{3} + 83718t^{2} + 12627t - 1058 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5)

|λ| = 10 λ ≈ 10; multiplicity 1; Jordan blocks 1 λ ≈ -8.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ 3.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ 3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 2; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
9
Spectral radius
$10$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10$11
$-8.09017$21, 1
$-6$11
$3.09017$21, 1
$3$21, 1
$2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda - 10\right) \left(\lambda - 2\right) \left(\lambda - 3\right)^{2} \left(\lambda^{2} + 5 \lambda - 25\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda - 10\right) \left(\lambda - 2\right) \left(\lambda - 3\right) \left(\lambda^{2} + 5 \lambda - 25\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 0 & 0 & 0 & 0 & 0 & 20 \\ 1 & 0 & 0 & 14 & 4 & -6 & -6 & 3 & 0 \\ 1 & 0 & 0 & -6 & 4 & 14 & 9 & 3 & 0 \\ 0 & 1 & 0 & 1 & 0 & 3 & 4 & 4 & 0 \\ 0 & 3 & 3 & -3 & 0 & -3 & 2 & 4 & 0 \\ 0 & 0 & 1 & 3 & 0 & 1 & 0 & 4 & 0 \\ 0 & 0 & 0 & 5 & 0 & -5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 5 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & 5 & 0\end{pmatrix}\notag\end{equation}$$