Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-384-81-c

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1
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
The bundle is zero: this presentation is the ambient variety itself.
ZL62 label
1111
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;2) \\ \det\mathcal{E} &\cong \mathcal{O}(0;0;0;0) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 81
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 384
$\chi(X, \mathrm{T}_X)$ = 12

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 384
$\mathrm{c}_1^2\mathrm{c}_2$ = 192
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 64
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 6 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 12 0 12
$\mathrm{HH}^{ 2 }$ = 54 0 0 54
$\mathrm{HH}^{ 3 }$ = 108 0 0 0 108
$\mathrm{HH}^{ 4 }$ = 81 0 0 0 0 81
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 11t + 11t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(363)
Coates et al. (2020) P1 x P1 x P1 x P1
Coates–Kasprzyk–Prince (2015) CKP(440)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 4t - 1 \right) \left( 4t + 1 \right) \left( 8t - 1 \right) \left( 8t + 1 \right)D^{4} + 64 t ^{2} \left( 128t^{2} - 5 \right)D^{3} + 16 t ^{2} \left( 1472t^{2} - 33 \right)D^{2} + 32 t ^{2} \left( 896t^{2} - 13 \right)D + 128 t ^{2} \left( 96t^{2} - 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1

|λ| = 8 λ ≈ -8; multiplicity 1; Jordan blocks 1 λ ≈ 8; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ 4; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ 0; multiplicity 6; Jordan blocks 1, 1, 1, 1, 1, 1 4 4 6
Sector dimension
16
Spectral radius
$8$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-8$11
$8$11
$-4$41, 1, 1, 1
$4$41, 1, 1, 1
$0$61, 1, 1, 1, 1, 1

Characteristic polynomial

$$\begin{equation}\lambda^{6} \left(\lambda + 4\right)^{4} \left(\lambda + 8\right) \left(\lambda - 4\right)^{4} \left(\lambda - 8\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right) \left(\lambda + 8\right) \left(\lambda - 4\right) \left(\lambda - 8\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 \\ 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 \\ 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$