Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-90-25-a

·

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

Defining data
Ambient
ℙ2P2 × 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_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w1 + w3 + w5)
Bundle
2 copies of O_1(1) box O_2(1) box O_3(1)
ZL62 label
202020.x4500x4500
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 25
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 90
$\chi(X, \mathrm{T}_X)$ = -26

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 90
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 135
$\mathrm{c}_1\mathrm{c}_3$ = 18
$\mathrm{c}_4$ = 45
Hodge diamond
1
00
0 3 0
0 0 0 0
0 2 33 2 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 26 0 26 x0
$\mathrm{HH}^{ 3 }$ = x0 + 10 0 0 x0 + 6 4
$\mathrm{HH}^{ 4 }$ = 28 0 0 3 0 25
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 20t + 48t^{2} + 20t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(605)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(623)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 24t - 1 \right) \left( 6t + 5 \right) ^{2} \left( 3t + 1 \right) ^{3} \left( 36t^{2} + 6t + 1 \right)D^{4} + 6 t \left( 6t + 5 \right) \left( 3t + 1 \right) ^{2} \left( 25920t^{4} + 31968t^{3} + 7560t^{2} + 708t + 17 \right)D^{3} + 3 t \left( 3t + 1 \right) \left( 3265920t^{6} + 7702128t^{5} + 6068952t^{4} + 1821096t^{3} + 228186t^{2} + 12073t + 110 \right)D^{2} + 3 t \left( 13996800t^{7} + 37278144t^{6} + 35497440t^{5} + 14527080t^{4} + 2752488t^{3} + 239220t^{2} + 8720t + 25 \right)D + 36 t ^{2} \left( 559872t^{6} + 1481328t^{5} + 1383048t^{4} + 532260t^{3} + 90582t^{2} + 6705t + 200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ2P2

|λ| = 24 λ ≈ 24; multiplicity 1; Jordan blocks 1 λ ≈ -3 - 5.196152i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -3 + 5.196152i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -3; multiplicity 7; Jordan blocks 1, 1, 1, 1, 3 3 3 7
Sector dimension
14
Spectral radius
$24$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$24$11
$-3 - 5.196152\mathrm{i}$31, 1, 1
$-3 + 5.196152\mathrm{i}$31, 1, 1
$-3$71, 1, 1, 1, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{7} \left(\lambda - 24\right) \left(\lambda^{2} + 6 \lambda + 36\right)^{3}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right)^{3} \left(\lambda - 24\right) \left(\lambda^{2} + 6 \lambda + 36\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 12 & 12 & 90 & 90 & 54 & 90 & 54 & 54 & 648 & 324 & 324 & 1260 \\ 1 & -1 & 2 & 2 & 10 & 10 & 4 & 16 & 10 & 10 & 108 & 54 & 54 & 216 \\ 1 & 2 & -1 & 2 & 10 & 16 & 10 & 10 & 4 & 10 & 108 & 54 & 54 & 216 \\ 1 & 2 & 2 & -1 & 16 & 10 & 10 & 10 & 10 & 4 & 108 & 54 & 54 & 216 \\ 0 & 1 & 1 & 0 & -2 & 2 & 0 & 2 & 0 & 2 & 8 & 2 & 2 & 24 \\ 0 & 1 & 0 & 1 & 2 & -2 & 0 & 2 & 2 & 0 & 8 & 6 & 4 & 24 \\ 0 & 1 & 0 & 0 & 2 & 2 & -1 & 0 & 1 & 1 & 16 & 8 & 6 & 30 \\ 0 & 0 & 1 & 1 & 2 & 2 & 2 & -2 & 0 & 0 & 8 & 4 & 6 & 24 \\ 0 & 0 & 1 & 0 & 2 & 0 & 1 & 2 & -1 & 1 & 16 & 6 & 8 & 30 \\ 0 & 0 & 0 & 1 & 0 & 2 & 1 & 2 & 1 & -1 & 16 & 10 & 10 & 30 \\ 0 & 0 & 0 & 0 & 1 & \frac{5}{2} & \frac{1}{2} & \frac{5}{2} & \frac{1}{2} & 2 & 3 & 3 & 3 & 12 \\ 0 & 0 & 0 & 0 & 1 & -2 & -1 & 1 & 2 & -1 & 0 & -3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 2 & -2 & -1 & -1 & 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{3}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$