Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-70-21-b

·

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
3 copies of E^(w1 + w4)
Bundle
3 copies of O(1,1)
ZL62 label
48.UUU
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 70
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 70
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 150
$\mathrm{c}_1\mathrm{c}_3$ = 0
$\mathrm{c}_4$ = 60
Hodge diamond
1
00
0 2 0
0 0 0 0
0 4 46 4 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 + 36 0 36 x0
$\mathrm{HH}^{ 3 }$ = x0 + 8 0 0 x0 + 4 4
$\mathrm{HH}^{ 4 }$ = 31 0 0 10 0 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 36t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(658)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(665)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 30t - 1 \right) \left( 10t + 7 \right) ^{2} \left( 2t + 1 \right) ^{3} \left( 25t^{2} + 15t + 1 \right)D^{4} + 2 t \left( 10t + 7 \right) \left( 2t + 1 \right) ^{2} \left( 75000t^{4} + 112000t^{3} + 50650t^{2} + 6750t + 73 \right)D^{3} + t \left( 2t + 1 \right) \left( 10500000t^{6} + 26230000t^{5} + 24744000t^{4} + 10814000t^{3} + 2128830t^{2} + 144429t + 658 \right)D^{2} + t \left( 30000000t^{7} + 85960000t^{6} + 97296000t^{5} + 54823600t^{4} + 15854920t^{3} + 2138388t^{2} + 95144t + 147 \right)D + 120 \left( 2t + 1 \right) \left( 30t + 7 \right) t ^{2} \left( 2000t^{4} + 4100t^{3} + 2810t^{2} + 673t + 28 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 4, 5)

|λ| = 30 λ ≈ 30; multiplicity 1; Jordan blocks 1 λ ≈ -13.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ -1.90983; multiplicity 2; Jordan blocks 1, 1 2 4 2
Sector dimension
9
Spectral radius
$30$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$30$11
$-13.09017$21, 1
$-2$41, 3
$-1.90983$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda - 30\right) \left(\lambda^{2} + 15 \lambda + 25\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda - 30\right) \left(\lambda^{2} + 15 \lambda + 25\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 30 & 30 & 300 & 240 & 240 & 2720 & 2720 & 1680 \\ 1 & -1 & 4 & 56 & 54 & 34 & 390 & 360 & 272 \\ 1 & 4 & -1 & 56 & 34 & 54 & 360 & 390 & 272 \\ 0 & 1 & 1 & -2 & 0 & 0 & 40 & 40 & 18 \\ 0 & 1 & 0 & 4 & -1 & 6 & 36 & 44 & 24 \\ 0 & 0 & 1 & 4 & 6 & -1 & 44 & 36 & 24 \\ 0 & 0 & 0 & 1 & -\frac{1}{2} & 2 & -1 & 4 & 3 \\ 0 & 0 & 0 & 1 & 2 & -\frac{1}{2} & 4 & -1 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 10 & 10 & 0\end{pmatrix}\notag\end{equation}$$