Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-320-69-b

·

No geometric description is recorded for this family (yet).

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(2w5) + E^(w3 + w5) + E^(w1 + w4)
Bundle
O_3(2) + O_2(1) box O_3(1) + O_1(1) box Q_2
ZL62 label
202040.CwEw2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 69
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 320
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 320
$\mathrm{c}_1^2\mathrm{c}_2$ = 176
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 10
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 4 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 4 0 x0 - 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 56 0 0 x1 - 32 88
$\mathrm{HH}^{ 4 }$ = 69 0 0 0 0 69
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 9t + 9t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(258)
Coates et al. (2020) MW(8)
Coates–Kasprzyk–Prince (2015) CKP(326)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t - 1 \right) \left( 2t + 1 \right) \left( 8000t^{6} + 528t^{4} + 60t^{2} - 1 \right) \left( 64404500t^{8} - 1791160t^{6} + 729408t^{4} + 2144t^{2} + 5 \right)D^{6} + 4 \left( 10304720000000t^{16} - 1452885248000t^{14} + 158410065280t^{12} - 11837396096t^{10} - 726357052t^{8} + 2787020t^{6} - 1531744t^{4} - 3536t^{2} - 5 \right)D^{5} + 4 \left( 79346344000000t^{16} - 7715142848000t^{14} + 1509213811040t^{12} - 96608493568t^{10} - 1083561820t^{8} + 48291628t^{6} + 2808508t^{4} + 5602t^{2} + 5 \right)D^{4} + 32 t ^{2} \left( 37612228000000t^{14} - 2748389976000t^{12} + 827208001440t^{10} - 40569448408t^{8} - 339211968t^{6} + 5857560t^{4} - 32366t^{2} - 51 \right)D^{3} + 32 t ^{2} \left( 74258388500000t^{14} - 4572988268000t^{12} + 1872888803480t^{10} - 62657669272t^{8} - 239219588t^{6} - 13755238t^{4} - 70290t^{2} - 15 \right)D^{2} + 512 t ^{4} \left( 4524416125000t^{12} - 258077892500t^{10} + 128903007200t^{8} - 2846763420t^{6} - 5734916t^{4} - 1401952t^{2} - 4225 \right)D + 3072 t ^{4} \left( 281769687500t^{12} - 15710668750t^{10} + 8827474000t^{8} - 131446365t^{6} - 200152t^{4} - 95918t^{2} - 245 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ4P4

|λ| = 8.323876 λ ≈ -8.323876; multiplicity 1; Jordan blocks 1 λ ≈ 8.323876; multiplicity 1; Jordan blocks 1 λ ≈ -1.746691 - 2.773876i; multiplicity 1; Jordan blocks 1 λ ≈ -1.746691 + 2.773876i; multiplicity 1; Jordan blocks 1 λ ≈ 1.746691 - 2.773876i; multiplicity 1; Jordan blocks 1 λ ≈ 1.746691 + 2.773876i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ 2; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$8.32387637$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-8.323876$11
$8.323876$11
$-1.746691 - 2.773876\mathrm{i}$11
$-1.746691 + 2.773876\mathrm{i}$11
$1.746691 - 2.773876\mathrm{i}$11
$1.746691 + 2.773876\mathrm{i}$11
$-2$11
$2$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{6} - 60 \lambda^{4} - 528 \lambda^{2} - 8000\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{6} - 60 \lambda^{4} - 528 \lambda^{2} - 8000\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 4 & 0 & 0 & 0 & 8 & 8 & 0 \\ 2 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 & 2 & 4 & 0 & 0 & 8 \\ 0 & 2 & 0 & 0 & 0 & 0 & 4 & 4 & 0 \\ 0 & 2 & 2 & 0 & 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 2 & 2 & -2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 2 & 4 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 4 & 0\end{pmatrix}\notag\end{equation}$$