Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-40-15-c

·

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

Defining data
Ambient
ℙ3P3 × ℙ3P3
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + w4) + E^(2w1 + 2w4)
Bundle
O_1(1) box O_2(1) + O_1(2) box O_2(2)
ZL62 label
3030.ix328
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 40
$\chi(X, \mathrm{T}_X)$ = -68

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 40
$\mathrm{c}_1^2\mathrm{c}_2$ = 88
$\mathrm{c}_2^2$ = 200
$\mathrm{c}_1\mathrm{c}_3$ = -40
$\mathrm{c}_4$ = 152
Hodge diamond
1
00
0 2 0
0 0 0 0
0 16 114 16 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 68 0 68 x0
$\mathrm{HH}^{ 3 }$ = x0 + 2 0 0 x0 + 1 1
$\mathrm{HH}^{ 4 }$ = 84 0 0 69 0 15
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 1 0 1 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 18t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(721)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(715)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 20t + 1 \right) \left( 60t - 1 \right) \left( 4t + 1 \right) ^{4}D^{4} + 8 t \left( 60t + 7 \right) \left( 100t + 1 \right) \left( 4t + 1 \right) ^{3}D^{3} + 4 t \left( 4t + 1 \right) ^{2} \left( 168000t^{3} + 44480t^{2} + 2368t + 9 \right)D^{2} + 8 t \left( 4t + 1 \right) \left( 480000t^{4} + 210560t^{3} + 26064t^{2} + 772t + 1 \right)D + 64 t ^{2} \left( 115200t^{4} + 73920t^{3} + 15904t^{2} + 1247t + 23 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ3P3

|λ| = 60 λ ≈ 60; multiplicity 1; Jordan blocks 1 λ ≈ -20; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 6; Jordan blocks 1, 2, 3 2 6
Sector dimension
9
Spectral radius
$60$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$60$11
$-20$21, 1
$-4$61, 2, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{6} \left(\lambda + 20\right)^{2} \left(\lambda - 60\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda + 20\right) \left(\lambda - 60\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 92 & 92 & 2112 & 1536 & 1536 & 33888 & 33888 & 117760 \\ 1 & 0 & 8 & 220 & 176 & 136 & 3072 & 2976 & 11296 \\ 1 & 8 & 0 & 220 & 136 & 176 & 2976 & 3072 & 11296 \\ 0 & 1 & 1 & 0 & 4 & 4 & 192 & 192 & 576 \\ 0 & 1 & 0 & 8 & -2 & 10 & 128 & 148 & 480 \\ 0 & 0 & 1 & 8 & 10 & -2 & 148 & 128 & 480 \\ 0 & 0 & 0 & 1 & -\frac{1}{3} & \frac{5}{3} & 0 & 8 & \frac{92}{3} \\ 0 & 0 & 0 & 1 & \frac{5}{3} & -\frac{1}{3} & 8 & 0 & \frac{92}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$