Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-486-100-a

·

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

Varieties in this family can be blown up to

Defining data
Ambient
ℙ2P2 × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
The bundle is zero: this presentation is the ambient variety itself.
ZL62 label
2020
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(0;0) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 3
$\mathrm{h}^0(X, \omega_X^\vee)$ = 100
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 486
$\chi(X, \mathrm{T}_X)$ = 16

$\omega_X^\vee \cong \mathcal{O}_X(3)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 486
$\mathrm{c}_1^2\mathrm{c}_2$ = 216
$\mathrm{c}_2^2$ = 99
$\mathrm{c}_1\mathrm{c}_3$ = 54
$\mathrm{c}_4$ = 9
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 3 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 16 0 16
$\mathrm{HH}^{ 2 }$ = 84 0 0 84
$\mathrm{HH}^{ 3 }$ = 160 0 0 0 160
$\mathrm{HH}^{ 4 }$ = 100 0 0 0 0 100
$\mathrm{HH}^{ 5 }$ = 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

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

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(14)
Coates et al. (2020) P2 x P2
Coates–Kasprzyk–Prince (2015) CKP(20)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 3t + 1 \right) \left( 6t - 1 \right) \left( 9t^{2} - 3t + 1 \right) \left( 36t^{2} + 6t + 1 \right)D^{4} + 162 t ^{3} \left( 432t^{3} + 7 \right)D^{3} + 27 t ^{3} \left( 10584t^{3} + 95 \right)D^{2} + 1296 t ^{3} \left( 351t^{3} + 2 \right)D + 972 t ^{3} \left( 240t^{3} + 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2

|λ| = 6 λ ≈ -3 - 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ -3 + 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ 6; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 2 2 2
Sector dimension
9
Spectral radius
$6$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-3 - 5.196152\mathrm{i}$11
$-3 + 5.196152\mathrm{i}$11
$6$11
$-3$21, 1
$1.5 - 2.598076\mathrm{i}$21, 1
$1.5 + 2.598076\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda - 6\right) \left(\lambda^{2} + 6 \lambda + 36\right) \left(\lambda^{2} - 3 \lambda + 9\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda - 6\right) \left(\lambda^{2} + 6 \lambda + 36\right) \left(\lambda^{2} - 3 \lambda + 9\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 3 & 3 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 3 & 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$