Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-160-39-a

·

An ordinary Gushel–Mukai fourfold, a hyperplane and quadric section of $\operatorname{Gr}(2,5)$.

This is a Fano fourfold of K3 type, appearing as case X10 in Bernardara–Fatighenti–Manivel–Tanturri.

Varieties in this family can be blown up to

  • 2-115-30-a (from a member containing a quadric surface)
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^(2w2)
Bundle
O(1) + O(2)
ZL62 label
41.15
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5) \\ \det\mathcal{E} &\cong \mathcal{O}(3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 160
$\chi(X, \mathrm{T}_X)$ = -24

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 160
$\mathrm{c}_1^2\mathrm{c}_2$ = 136
$\mathrm{c}_2^2$ = 116
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 28
Hodge diamond
1
00
0 1 0
0 0 0 0
0 1 22 1 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 + x1 + 24 0 x0 + 24 x1
$\mathrm{HH}^{ 3 }$ = x1 + 37 0 0 x1 + 1 36
$\mathrm{HH}^{ 4 }$ = 40 0 0 1 0 39
$\mathrm{HH}^{ 5 }$ = 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

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

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(496)
Coates et al. (2020) V(4,10)
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 256t^{4} + 176t^{2} - 1 \right)D^{4} + 64 t ^{2} \left( 32t^{2} + 11 \right)D^{3} + 16 t ^{2} \left( 352t^{2} + 67 \right)D^{2} + 32 t ^{2} \left( 192t^{2} + 23 \right)D + 192 t ^{2} \left( 12t^{2} + 1 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 5)

|λ| = 13.32076 λ ≈ -13.32076; multiplicity 1; Jordan blocks 1 λ ≈ 13.32076; multiplicity 1; Jordan blocks 1 λ ≈ 0 - 1.201132i; multiplicity 1; Jordan blocks 1 λ ≈ 0 + 1.201132i; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 2
Sector dimension
6
Spectral radius
$13.3207627$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-13.32076$11
$13.32076$11
$0 - 1.201132\mathrm{i}$11
$0 + 1.201132\mathrm{i}$11
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda^{4} - 176 \lambda^{2} - 256\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{4} - 176 \lambda^{2} - 256\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 0 & 0 & 96 & 0 \\ 2 & 0 & 12 & 20 & 0 & 48 \\ 0 & 2 & 0 & 0 & 8 & 0 \\ 0 & 2 & 0 & 0 & 16 & 0 \\ 0 & 0 & 2 & 3 & 0 & 6 \\ 0 & 0 & 0 & 0 & 4 & 0\end{pmatrix}\notag\end{equation}$$