Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-20-10-a

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An intersection of two quadrics in $\operatorname{Gr}(2,5)$, Küchle’s fourfold of type (b2).

Defining data
Ambient
Gr(2, 5)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
2 copies of E^(2w2)
Bundle
2 copies of O(2)
ZL62 label
41.55
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5) \\ \det\mathcal{E} &\cong \mathcal{O}(4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 10
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 20
$\chi(X, \mathrm{T}_X)$ = -72

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 20
$\mathrm{c}_1^2\mathrm{c}_2$ = 68
$\mathrm{c}_2^2$ = 232
$\mathrm{c}_1\mathrm{c}_3$ = -52
$\mathrm{c}_4$ = 176
Hodge diamond
1
00
0 1 0
0 0 0 0
0 20 132 20 0
0 0 0 0
0 1 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 72 0 72 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 101 0 0 91 0 10
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 2 0 2 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 5t + 8t^{2} + 5t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(740)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 12t + 1 \right) ^{4} \left( 2224t^{2} + 152t - 1 \right)D^{4} + 8 t \left( 12t + 1 \right) ^{3} \left( 33360t^{2} + 2936t + 29 \right)D^{3} + 4 t \left( 12t + 1 \right) ^{2} \left( 2802240t^{3} + 352704t^{2} + 10792t + 37 \right)D^{2} + 32 t \left( 12t + 1 \right) \left( 6004800t^{4} + 1070496t^{3} + 59328t^{2} + 993t + 1 \right)D + 576 t ^{2} \left( 1921536t^{4} + 465984t^{3} + 39856t^{2} + 1377t + 15 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 5)

|λ| = 165.4427 λ ≈ 165.4427; multiplicity 1; Jordan blocks 1 λ ≈ -13.44272; multiplicity 1; Jordan blocks 1 λ ≈ -12; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
6
Spectral radius
$165.442719$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$165.4427$11
$-13.44272$11
$-12$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 12\right)^{4} \left(\lambda^{2} - 152 \lambda - 2224\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 12\right)^{3} \left(\lambda^{2} - 152 \lambda - 2224\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 1080 & 46080 & 71424 & 3301632 & 39398400 \\ 1 & 32 & 1848 & 2872 & 135552 & 1650816 \\ 0 & 1 & 8 & 32 & 1648 & 20736 \\ 0 & 1 & 28 & 32 & 2048 & 25344 \\ 0 & 0 & 1 & \frac{3}{2} & 32 & 540 \\ 0 & 0 & 0 & 0 & 2 & 0\end{pmatrix}\notag\end{equation}$$