Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-42-15-b

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A codimension-five linear section of $\operatorname{Gr}(3,6)$, Küchle’s fourfold of type (c1).

Defining data
Ambient
Gr(3, 6)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
5 copies of E^(w3)
Bundle
5 copies of O(1)
ZL62 label
53.22222
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 42
$\chi(X, \mathrm{T}_X)$ = -40

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 42
$\mathrm{c}_1^2\mathrm{c}_2$ = 84
$\mathrm{c}_2^2$ = 168
$\mathrm{c}_1\mathrm{c}_3$ = -12
$\mathrm{c}_4$ = 66
Hodge diamond
1
00
0 1 0
0 0 0 0
0 5 52 5 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 + 40 0 40 x0
$\mathrm{HH}^{ 3 }$ = x0 + 5 0 0 x0 + 5 0
$\mathrm{HH}^{ 4 }$ = 30 0 0 15 0 15
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 20t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) No match
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 5t + 1 \right) \left( 58t - 1 \right) \left( 6t + 1 \right) ^{3}D^{4} + 2 t \left( 6t + 1 \right) ^{2} \left( 8700t^{2} + 1852t + 35 \right)D^{3} + t \left( 6t + 1 \right) \left( 365400t^{3} + 103044t^{2} + 6850t + 45 \right)D^{2} + 2 t \left( 1566000t^{4} + 598752t^{3} + 71388t^{2} + 2576t + 5 \right)D + 144 t ^{2} \left( 10440t^{3} + 3564t^{2} + 363t + 10 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(3, 6)

|λ| = 58 λ ≈ 58; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 3; Jordan blocks 3 λ ≈ -5; multiplicity 2; Jordan blocks 1, 1 3 2
Sector dimension
6
Spectral radius
$58$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$58$11
$-6$33
$-5$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{2} \left(\lambda + 6\right)^{3} \left(\lambda - 58\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda + 6\right)^{3} \left(\lambda - 58\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 180 & 3402 & 3402 & 21960 & 241200 \\ 1 & 10 & 294 & 294 & 1950 & 21960 \\ 0 & 1 & 5 & 10 & 70 & 810 \\ 0 & 1 & 10 & 5 & 70 & 810 \\ 0 & 0 & \frac{21}{5} & \frac{21}{5} & 10 & 180 \\ 0 & 0 & 0 & 0 & 1 & 0\end{pmatrix}\notag\end{equation}$$