Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-320-69-d

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

Varieties in this family can be blown up to

Defining data
Ambient
Gr(2, 6)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w2) + E^(2w1)
Bundle
O(1) + Sym^2 U^v
ZL62 label
51.15
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 69
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 320
$\chi(X, \mathrm{T}_X)$ = 2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 320
$\mathrm{c}_1^2\mathrm{c}_2$ = 176
$\mathrm{c}_2^2$ = 100
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 6 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 - 2 0 x0 - 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 56 0 0 x1 - 34 90
$\mathrm{HH}^{ 4 }$ = 69 0 0 0 0 69
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 9t + 9t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(139)
Coates et al. (2020) MW(7)
Coates–Kasprzyk–Prince (2015) CKP(189)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 8t - 1 \right) \left( 8t + 1 \right) \left( 16t^{2} + 1 \right)D^{4} + 64 t ^{2} \left( 128t^{2} + 3 \right)D^{3} + 16 t ^{2} \left( 1456t^{2} + 19 \right)D^{2} + 32 t ^{2} \left( 864t^{2} + 7 \right)D + 64 t ^{2} \left( 180t^{2} + 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 6)

|λ| = 8 λ ≈ -8; multiplicity 1; Jordan blocks 1 λ ≈ 8; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 4i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 4i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
6
Spectral radius
$8$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-8$11
$8$11
$0 - 4\mathrm{i}$11
$0 + 4\mathrm{i}$11
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 8\right) \left(\lambda - 8\right) \left(\lambda^{2} + 16\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 8\right) \left(\lambda - 8\right) \left(\lambda^{2} + 16\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 0 & 0 & 32 & 0 \\ 2 & 0 & 4 & 4 & 0 & 16 \\ 0 & 2 & 0 & 0 & 8 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 3 & 0 & 2 \\ 0 & 0 & 0 & 0 & 4 & 0\end{pmatrix}\notag\end{equation}$$