Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-15-9-a

·

A hyperplane and cubic section of $\operatorname{Gr}(2,5)$, Küchle’s fourfold of type (b1).

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
E^(w2) + E^(3w2)
Bundle
O(1) + O(3)
ZL62 label
41.19
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 9
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 15
$\chi(X, \mathrm{T}_X)$ = -109

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 15
$\mathrm{c}_1^2\mathrm{c}_2$ = 66
$\mathrm{c}_2^2$ = 291
$\mathrm{c}_1\mathrm{c}_3$ = -84
$\mathrm{c}_4$ = 318
Hodge diamond
1
00
0 1 0
0 0 0 0
0 41 232 41 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 + 109 0 109 x0
$\mathrm{HH}^{ 3 }$ = x0 + 1 0 0 x0 + 1 0
$\mathrm{HH}^{ 4 }$ = 200 0 0 191 0 9
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 9 0 9 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

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

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(742)
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( 18t + 1 \right) ^{4} \left( 5751t^{2} + 261t - 1 \right)D^{4} + 18 t \left( 18t + 1 \right) ^{3} \left( 57510t^{2} + 3366t + 23 \right)D^{3} + 3 t \left( 18t + 1 \right) ^{2} \left( 21738780t^{3} + 1821528t^{2} + 37251t + 88 \right)D^{2} + 3 t \left( 18t + 1 \right) \left( 558997200t^{4} + 66379824t^{3} + 2450412t^{2} + 27342t + 19 \right)D + 72 t ^{2} \left( 201238992t^{4} + 32516316t^{3} + 1852146t^{2} + 42579t + 308 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 5)

|λ| = 281.4346 λ ≈ 281.4346; multiplicity 1; Jordan blocks 1 λ ≈ -20.43459; multiplicity 1; Jordan blocks 1 λ ≈ -18; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
6
Spectral radius
$281.434588$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$281.4346$11
$-20.43459$11
$-18$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 18\right)^{4} \left(\lambda^{2} - 261 \lambda - 5751\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 18\right)^{3} \left(\lambda^{2} - 261 \lambda - 5751\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2772 & 200988 & 310284 & 23784192 & 430673760 \\ 1 & 57 & 5328 & 8244 & 644490 & 11892096 \\ 0 & 1 & 18 & 57 & 4824 & 91692 \\ 0 & 1 & 48 & 57 & 5832 & 109296 \\ 0 & 0 & 1 & \frac{3}{2} & 57 & 1386 \\ 0 & 0 & 0 & 0 & 2 & 0\end{pmatrix}\notag\end{equation}$$