Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-204-47-d

·

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

Defining data
Ambient
Fl(1, 2, 3) × Fl(1, 2, 3)
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
E^(w1 + w3) + E^(w2 + w4)
Bundle
O_1(1,0) box O_2(1,0) + O_1(0,1) box O_2(0,1)
ZL62 label
2222.WT
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,2;2,2) \\ \det\mathcal{E} &\cong \mathcal{O}(1,1;1,1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 47
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 204
$\chi(X, \mathrm{T}_X)$ = 0

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 204
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 102
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 0 x0 x1
$\mathrm{HH}^{ 3 }$ = x1 + 30 0 0 x1 - 2 32
$\mathrm{HH}^{ 4 }$ = 47 0 0 0 0 47
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 42t + 118t^{2} + 42t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(396)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(469)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 6t + 1 \right) \left( 12t - 1 \right) \left( 4t + 1 \right) ^{2} \left( 74t + 17 \right) ^{2} \left( 7t^{2} - t + 1 \right)D^{4} + 6 t \left( 4t + 1 \right) \left( 74t + 17 \right) \left( 248640t^{5} + 94000t^{4} + 24008t^{3} + 7676t^{2} + 1046t + 19 \right)D^{3} + t \left( 1545546240t^{7} + 1279217280t^{6} + 451468544t^{5} + 106767864t^{4} + 21206284t^{3} + 2875346t^{2} + 175353t + 1258 \right)D^{2} + t \left( 2207923200t^{7} + 1782251520t^{6} + 586028256t^{5} + 117789120t^{4} + 19426696t^{3} + 2348244t^{2} + 126480t + 289 \right)D + 8 t ^{2} \left( 132475392t^{6} + 105259968t^{5} + 33095608t^{4} + 5949636t^{3} + 847186t^{2} + 93993t + 4624 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3) × Fl(1, 2, 3)

|λ| = 12 λ ≈ 12; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 5; Jordan blocks 1, 1, 1, 2 λ ≈ 0.5 - 2.598076i; multiplicity 4; Jordan blocks 1, 1, 1, 1 λ ≈ 0.5 + 2.598076i; multiplicity 4; Jordan blocks 1, 1, 1, 1 2 5 4 4
Sector dimension
16
Spectral radius
$12$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12$11
$-6$21, 1
$-4$51, 1, 1, 2
$0.5 - 2.598076\mathrm{i}$41, 1, 1, 1
$0.5 + 2.598076\mathrm{i}$41, 1, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{5} \left(\lambda + 6\right)^{2} \left(\lambda - 12\right) \left(\lambda^{2} - \lambda + 7\right)^{4}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 6\right) \left(\lambda - 12\right) \left(\lambda^{2} - \lambda + 7\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 4 & 4 & 3 & 6 & 6 & 3 & 6 & 6 & 8 & 12 & 8 & 12 & 20 \\ 1 & -1 & 1 & 1 & 0 & 2 & 2 & 4 & 0 & 0 & 2 & 0 & 3 & 3 & 3 & 4 \\ 1 & 1 & -1 & 0 & 1 & 0 & 2 & 0 & 2 & 4 & 2 & 0 & 0 & 0 & 3 & 4 \\ 1 & 1 & 0 & -1 & 1 & 0 & 2 & 0 & 2 & 4 & 2 & 3 & 3 & 3 & 3 & 4 \\ 1 & 0 & 1 & 1 & -1 & 2 & 2 & 4 & 0 & 0 & 2 & 3 & 3 & 0 & 0 & 4 \\ 0 & 1 & 2 & -1 & -2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 \\ 0 & 1 & 0 & 3 & 1 & 0 & -1 & 2 & 0 & 2 & 0 & 2 & 4 & 2 & 2 & 3 \\ 0 & 1 & 0 & 0 & 1 & 1 & 1 & -2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & -2 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -2 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 3 & 0 & 0 & 2 & 0 & 2 & -1 & 2 & 2 & 2 & 4 & 3 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & -2 & 0 & -1 & 1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 5 & 1 & 1 & 7 & 2 & 1 & -1 & 1 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0 & 1 & -8 & -3 & -3 & -5 & 1 & -1 & -1 & -2 & -2 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 5 & 5 & 2 & 3 & 2 & 1 & 2 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 2 & 3 & 0\end{pmatrix}\notag\end{equation}$$