Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-80-23-a

·

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

Defining data
Ambient
Fl(1, 4, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)ω1ω2ω3ω4
Bundle
E^(2w1) + E^(w1 + w4) + E^(2w4)
Bundle
O(2,0) + O(1,1) + O(0,2)
ZL62 label
48.7U4
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4,4) \\ \det\mathcal{E} &\cong \mathcal{O}(3,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 23
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 80
$\chi(X, \mathrm{T}_X)$ = -26

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 80
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 30 2 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 + x2 + 26 0 x0 + 26 x2
$\mathrm{HH}^{ 3 }$ = x1 + 2 0 0 x1 2
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 15 0 0 x1 - x2 - 8 0 23
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 18t + 42t^{2} + 18t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(651)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(659)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 8t + 1 \right) \left( 28t - 1 \right) \left( 28t + 5 \right) ^{2} \left( 4t + 1 \right) ^{3}D^{4} + 8 t \left( 28t + 5 \right) \left( 4t + 1 \right) ^{2} \left( 25088t^{3} + 8848t^{2} + 904t + 17 \right)D^{3} + 8 t \left( 4t + 1 \right) \left( 8078336t^{5} + 5456640t^{4} + 1391488t^{3} + 160032t^{2} + 7412t + 55 \right)D^{2} + 4 t \left( 78675968t^{6} + 67587072t^{5} + 22925056t^{4} + 3831296t^{3} + 314880t^{2} + 10260t + 25 \right)D + 64 t ^{2} \left( 2107392t^{5} + 1724800t^{4} + 552944t^{3} + 86192t^{2} + 6435t + 175 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 5)

|λ| = 28 λ ≈ 28; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2 4 2
Sector dimension
9
Spectral radius
$28$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28$11
$-8$21, 1
$-4$41, 3
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right)^{4} \left(\lambda + 8\right)^{2} \left(\lambda - 28\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right)^{3} \left(\lambda + 8\right) \left(\lambda - 28\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 28 & 28 & 288 & 192 & 192 & 2016 & 2016 & 4480 \\ 1 & 0 & 4 & 44 & 32 & 24 & 288 & 288 & 672 \\ 1 & 4 & 0 & 44 & 24 & 32 & 288 & 288 & 672 \\ 0 & 1 & 1 & 0 & 4 & 4 & 48 & 48 & 96 \\ 0 & 1 & 0 & 4 & -2 & 2 & 16 & 20 & 48 \\ 0 & 0 & 1 & 4 & 2 & -2 & 20 & 16 & 48 \\ 0 & 0 & 0 & 1 & -\frac{1}{3} & \frac{5}{3} & 0 & 4 & \frac{28}{3} \\ 0 & 0 & 0 & 1 & \frac{5}{3} & -\frac{1}{3} & 4 & 0 & \frac{28}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$