Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-28-12-a

·

A quadric section of a codimension-three linear section of $\operatorname{Gr}(2,6)$, Küchle’s fourfold of type (b6).

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
3 copies of E^(w2) + E^(2w2)
Bundle
3 copies of O(1) + O(2)
ZL62 label
51.1116
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 12
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 28
$\chi(X, \mathrm{T}_X)$ = -63

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 28
$\mathrm{c}_1^2\mathrm{c}_2$ = 76
$\mathrm{c}_2^2$ = 208
$\mathrm{c}_1\mathrm{c}_3$ = -40
$\mathrm{c}_4$ = 140
Hodge diamond
1
00
0 1 0
0 0 0 0
0 15 106 15 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
$\mathrm{HH}^{ 2 }$ = x0 + 63 0 63 x0
$\mathrm{HH}^{ 3 }$ = x0 + 3 0 0 x0 + 3 0
$\mathrm{HH}^{ 4 }$ = 78 0 0 66 0 12
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 7t + 12t^{2} + 7t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(734)
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 12t + 1 \right) \left( 100t - 1 \right) \left( 8t + 1 \right) ^{4}D^{4} + 64 t \left( 8t + 1 \right) ^{3} \left( 1500t^{2} + 163t + 2 \right)D^{3} + 2 t \left( 8t + 1 \right) ^{2} \left( 1344000t^{3} + 227968t^{2} + 9258t + 41 \right)D^{2} + 2 t \left( 8t + 1 \right) \left( 15360000t^{4} + 3856384t^{3} + 296112t^{2} + 6692t + 9 \right)D + 32 t ^{2} \left( 3686400t^{4} + 1287168t^{3} + 156816t^{2} + 7583t + 112 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 6)

|λ| = 100 λ ≈ 100; multiplicity 1; Jordan blocks 1 λ ≈ -12; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
6
Spectral radius
$100$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$100$11
$-12$11
$-8$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 8\right)^{4} \left(\lambda + 12\right) \left(\lambda - 100\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 8\right)^{3} \left(\lambda + 12\right) \left(\lambda - 100\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 448 & 10440 & 19368 & 479360 & 4650240 \\ 1 & 18 & 620 & 1152 & 29040 & 287616 \\ 0 & 1 & -2 & 12 & 440 & 4536 \\ 0 & 1 & 16 & 22 & 740 & 7416 \\ 0 & 0 & 1 & \frac{9}{5} & 18 & \frac{1344}{5} \\ 0 & 0 & 0 & 0 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$