Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-224-51-a

·

The genus-eight Mukai fourfold, a codimension-four linear section of $\operatorname{Gr}(2,6)$.

Varieties in this family can be blown up to

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
4 copies of E^(w2)
Bundle
4 copies of O(1)
ZL62 label
51.1111
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6) \\ \det\mathcal{E} &\cong \mathcal{O}(4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 51
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 224
$\chi(X, \mathrm{T}_X)$ = -9

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 224
$\mathrm{c}_1^2\mathrm{c}_2$ = 152
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 1 0
0 0 0 0
0 0 8 0 0
0 0 0 0
0 1 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + 15 0 x0 + 9 6
$\mathrm{HH}^{ 3 }$ = 55 0 0 0 55
$\mathrm{HH}^{ 4 }$ = 51 0 0 0 0 51
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 6t + 6t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(365)
Coates et al. (2020) V(4,14)
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 4t^{2} + 1 \right) \left( 108t^{2} - 1 \right)D^{4} + 32 t ^{2} \left( 108t^{2} + 13 \right)D^{3} + 24 t ^{2} \left( 406t^{2} + 27 \right)D^{2} + 16 t ^{2} \left( 708t^{2} + 29 \right)D + 128 t ^{2} \left( 36t^{2} + 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 6)

|λ| = 10.3923 λ ≈ -10.3923; multiplicity 1; Jordan blocks 1 λ ≈ 10.3923; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
6
Spectral radius
$10.3923048$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-10.3923$11
$10.3923$11
$0 - 2\mathrm{i}$11
$0 + 2\mathrm{i}$11
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda^{2} + 4\right) \left(\lambda^{2} - 108\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{2} + 4\right) \left(\lambda^{2} - 108\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 0 & 0 & 40 & 0 \\ 2 & 0 & 6 & 12 & 0 & 24 \\ 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 10 & 0 \\ 0 & 0 & 2 & \frac{18}{5} & 0 & \frac{24}{5} \\ 0 & 0 & 0 & 0 & \frac{10}{3} & 0\end{pmatrix}\notag\end{equation}$$