Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 1-128-33-a

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A smooth complete intersection of three quadrics in $\mathbb P^7$.

Defining data
Ambient
ℙ7P7
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
3 copies of E^(2w1)
Bundle
3 copies of O(2)
ZL62 label
700.777
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(8) \\ \det\mathcal{E} &\cong \mathcal{O}(6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 33
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 128
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 128
$\mathrm{c}_1^2\mathrm{c}_2$ = 128
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 0
$\mathrm{c}_4$ = 48
Hodge diamond
1
00
0 1 0
0 0 0 0
0 3 38 3 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 + 36 0 36 x0
$\mathrm{HH}^{ 3 }$ = x0 + 28 0 0 x0 28
$\mathrm{HH}^{ 4 }$ = 39 0 0 6 0 33
$\mathrm{HH}^{ 5 }$ = 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

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

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(568)
Coates et al. (2020) V(4,8)
Coates–Kasprzyk–Prince (2015) CKP(596)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 16t - 1 \right) \left( 16t + 1 \right)D^{4} + 1024 t ^{2}D^{3} + 1536 t ^{2}D^{2} + 1024 t ^{2}D + 256 t ^{2} \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ7P7

|λ| = 16 λ ≈ -16; multiplicity 1; Jordan blocks 1 λ ≈ 16; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 3; Jordan blocks 3 3
Sector dimension
5
Spectral radius
$16$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-16$11
$16$11
$0$33

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 16\right) \left(\lambda - 16\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 16\right) \left(\lambda - 16\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 0 & 448 & 0 \\ 2 & 0 & 48 & 0 & 448 \\ 0 & 2 & 0 & 48 & 0 \\ 0 & 0 & 2 & 0 & 16 \\ 0 & 0 & 0 & 2 & 0\end{pmatrix}\notag\end{equation}$$