Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 5-120-31-a

·

This is a Fano fourfold of K3 type, appearing as case K3-60 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5 (marked)ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2 + w3 + w4 + w5)
Bundle
O_1(1) box O_2(1) box O_3(1) box O_4(1) box O_5(1)
ZL62 label
11111.x6000
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;2;2) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;1;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 31
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 120
$\chi(X, \mathrm{T}_X)$ = -16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 120
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 5 0
0 0 0 0
0 1 26 1 0
0 0 0 0
0 5 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 16 0 16 x0
$\mathrm{HH}^{ 3 }$ = x0 + 9 0 0 x0 + 5 4
$\mathrm{HH}^{ 4 }$ = 32 0 0 1 0 31
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 26t + 66t^{2} + 26t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(610)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(628)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 4t - 1 \right) \left( 4t + 1 \right) \left( 20t - 1 \right) \left( 5t + 1 \right) ^{2}D^{4} + 4 t \left( 5t + 1 \right) \left( 4000t^{3} + 320t^{2} - 166t - 5 \right)D^{3} + t \left( 280000t^{4} + 64400t^{3} - 4820t^{2} - 1275t - 13 \right)D^{2} + t \left( 400000t^{4} + 78400t^{3} - 5280t^{2} - 1022t - 3 \right)D + 80 t ^{2} \left( 2400t^{3} + 420t^{2} - 27t - 4 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ1P1

|λ| = 20 λ ≈ 20; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 6; Jordan blocks 1, 1, 1, 1, 2 λ ≈ -4; multiplicity 10; Jordan blocks 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 λ ≈ 4; multiplicity 5; Jordan blocks 1, 1, 1, 1, 1 6 10 5
Sector dimension
22
Spectral radius
$20$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20$11
$-5$61, 1, 1, 1, 2
$-4$101, 1, 1, 1, 1, 1, 1, 1, 1, 1
$4$51, 1, 1, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{10} \left(\lambda + 5\right)^{6} \left(\lambda - 20\right) \left(\lambda - 4\right)^{5}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right) \left(\lambda + 5\right)^{2} \left(\lambda - 20\right) \left(\lambda - 4\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 8 & 8 & 8 & 8 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 48 & 48 & 48 & 48 & 48 & 120 \\ 1 & -1 & 1 & 1 & 1 & 1 & 6 & 6 & 6 & 6 & 2 & 2 & 2 & 2 & 2 & 2 & 12 & 12 & 12 & 12 & 6 & 24 \\ 1 & 1 & -1 & 1 & 1 & 1 & 6 & 2 & 2 & 2 & 6 & 6 & 6 & 2 & 2 & 2 & 12 & 12 & 12 & 6 & 12 & 24 \\ 1 & 1 & 1 & -1 & 1 & 1 & 2 & 6 & 2 & 2 & 6 & 2 & 2 & 6 & 6 & 2 & 12 & 6 & 6 & 12 & 12 & 24 \\ 1 & 1 & 1 & 1 & -1 & 1 & 2 & 2 & 6 & 2 & 2 & 6 & 2 & 6 & 2 & 6 & 6 & 12 & 6 & 12 & 12 & 24 \\ 1 & 1 & 1 & 1 & 1 & -1 & 2 & 2 & 2 & 6 & 2 & 2 & 6 & 2 & 6 & 6 & 6 & 6 & 12 & 6 & 6 & 24 \\ 0 & 1 & 1 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 4 & 4 & 4 & 2 & 2 & 6 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 4 & 2 & 2 & 4 & 2 & 6 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & -2 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 2 & 4 & 2 & 4 & 2 & 6 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 0 & 0 & 1 & 0 & 1 & 1 & 2 & 2 & 4 & 2 & 0 & 6 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 0 & 4 & 2 & 2 & 2 & 4 & 6 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & -2 & 1 & 1 & 0 & 1 & 2 & 4 & 2 & 2 & 4 & 6 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & -2 & 0 & 1 & 1 & 2 & 2 & 4 & 0 & 2 & 6 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & -2 & 1 & 1 & 2 & 2 & 0 & 4 & 4 & 6 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & -2 & 1 & 2 & 0 & 2 & 2 & 2 & 6 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & -2 & 0 & 2 & 2 & 2 & 2 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & -1 & -1 & -3 & -3 & -1 & -1 & -1 & -1 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & -1 & -3 & -1 & -1 & -3 & -1 & -1 & -1 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 3 & 1 & 1 & 3 & 1 & 3 & 3 & 3 & 3 & 1 & 3 & 3 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 2 & 2 & 0 & 2 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 0 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$