Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-162-39-b

·

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
2 copies of E^(w2) + 2 copies of E^(w3)
Bundle
2 copies of O(1,0) + 2 copies of O(0,1)
ZL62 label
45.2211
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 162
$\chi(X, \mathrm{T}_X)$ = -8

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 162
$\mathrm{c}_1^2\mathrm{c}_2$ = 132
$\mathrm{c}_2^2$ = 108
$\mathrm{c}_1\mathrm{c}_3$ = 44
$\mathrm{c}_4$ = 14
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 8 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 + 8 0 x0 + 8 x1
$\mathrm{HH}^{ 3 }$ = x1 + 34 0 0 x1 + 10 24
$\mathrm{HH}^{ 4 }$ = 39 0 0 0 0 39
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 92t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(464)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 14t - 1 \right) \left( 2t + 1 \right) ^{2} \left( 112t + 81 \right) ^{2} \left( 7t^{3} + 3t^{2} + 8t + 1 \right)D^{4} + 2 t \left( 2t + 1 \right) \left( 112t + 81 \right) \left( 109760t^{5} + 159544t^{4} + 141190t^{3} + 94033t^{2} + 20350t + 274 \right)D^{3} + t \left( 172103680t^{7} + 445098752t^{6} + 507806600t^{5} + 368844868t^{4} + 182426782t^{3} + 50135031t^{2} + 5246867t + 28755 \right)D^{2} + t \left( 245862400t^{7} + 620714752t^{6} + 654356976t^{5} + 413782544t^{4} + 181274680t^{3} + 44373744t^{2} + 3756942t + 6561 \right)D + 8 t ^{2} \left( 14751744t^{6} + 36681792t^{5} + 36791944t^{4} + 21044240t^{3} + 8411228t^{2} + 1887867t + 131220 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5)

|λ| = 14 λ ≈ 14; multiplicity 1; Jordan blocks 1 λ ≈ -7.729032; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 λ ≈ -0.1354842 - 0.9419768i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -0.1354842 + 0.9419768i; multiplicity 2; Jordan blocks 1, 1 2 3 2 2
Sector dimension
10
Spectral radius
$14$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$14$11
$-7.729032$21, 1
$-2$31, 2
$-0.1354842 - 0.9419768\mathrm{i}$21, 1
$-0.1354842 + 0.9419768\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda - 14\right) \left(\lambda^{3} + 8 \lambda^{2} + 3 \lambda + 7\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 14\right) \left(\lambda^{3} + 8 \lambda^{2} + 3 \lambda + 7\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 10 & 10 & 12 & 18 & 18 & 12 & 24 & 48 & 60 \\ 1 & -1 & 2 & 6 & 4 & 8 & 2 & 6 & 6 & 16 \\ 1 & 2 & -1 & 2 & 8 & 4 & 6 & 3 & 12 & 16 \\ 0 & 1 & 0 & -1 & 2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & -1 & 1 & 0 & 4 & 4 & 6 \\ 0 & 2 & 1 & 0 & 1 & -1 & 1 & 2 & 8 & 6 \\ 0 & 0 & 1 & 1 & 0 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & \frac{17}{3} & \frac{5}{3} & 4 & -1 & 4 & \frac{20}{3} \\ 0 & 0 & 0 & 2 & \frac{5}{6} & \frac{17}{6} & \frac{1}{2} & 1 & -1 & \frac{10}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 3 & 0\end{pmatrix}\notag\end{equation}$$