Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-106-28-a

·

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

Defining data
Ambient
Gr(2, 4) × Gr(2, 4)
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_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + w5) + E^(w2 + w4)
Bundle
U_1^v box O_2(1) + O_1(1) box U_2^v
ZL62 label
3131.mj
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 28
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 106
$\chi(X, \mathrm{T}_X)$ = -16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 106
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 123
$\mathrm{c}_1\mathrm{c}_3$ = 30
$\mathrm{c}_4$ = 21
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 15 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 + x2 + 16 0 x0 + 16 x2
$\mathrm{HH}^{ 3 }$ = x1 + 8 0 0 x1 8
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 12 0 0 x1 - x2 - 16 0 28
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 23t + 58t^{2} + 23t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(583)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t + 1 \right) \left( 3t + 1 \right) \left( 44t^{2} + 18t - 1 \right) \left( 104t^{2} + 257t + 53 \right) ^{2} \left( 19t^{3} + 22t^{2} + 11t + 1 \right)D^{4} + 2 t \left( 104t^{2} + 257t + 53 \right) \left( 2608320t^{8} + 12743312t^{7} + 21248560t^{6} + 18174656t^{5} + 8881477t^{4} + 2499762t^{3} + 382262t^{2} + 26602t + 363 \right)D^{3} + t \left( 1898856960t^{10} + 13988713856t^{9} + 38525786392t^{8} + 51418498136t^{7} + 39183964160t^{6} + 18266302215t^{5} + 5294445625t^{4} + 932078772t^{3} + 92151476t^{2} + 4135791t + 24857 \right)D^{2} + 2 t \left( 1356326400t^{10} + 10001271488t^{9} + 27171604200t^{8} + 34430497896t^{7} + 24386471976t^{6} + 10419445704t^{5} + 2736350704t^{4} + 430521495t^{3} + 37142241t^{2} + 1368407t + 2809 \right)D + 4 t ^{2} \left( 325518336t^{9} + 2401688640t^{8} + 6474423984t^{7} + 7921195888t^{6} + 5334111816t^{5} + 2143044057t^{4} + 523790167t^{3} + 75673629t^{2} + 5853267t + 179776 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Gr(2, 4) × Gr(2, 4)

|λ| = 20.18034 λ ≈ 20.18034; multiplicity 1; Jordan blocks 1 λ ≈ -8.729032; multiplicity 2; Jordan blocks 1, 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2.18034; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.135484 - 0.9419768i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.135484 + 0.9419768i; multiplicity 2; Jordan blocks 1, 1 2 2 2 2
Sector dimension
11
Spectral radius
$20.1803399$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20.18034$11
$-8.729032$21, 1
$-3$21, 1
$-2.18034$11
$-2$11
$-1.135484 - 0.9419768\mathrm{i}$21, 1
$-1.135484 + 0.9419768\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right)^{2} \left(\lambda^{2} - 18 \lambda - 44\right) \left(\lambda^{3} + 11 \lambda^{2} + 22 \lambda + 19\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda^{2} - 18 \lambda - 44\right) \left(\lambda^{3} + 11 \lambda^{2} + 22 \lambda + 19\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 16 & 48 & 108 & 39 & 48 & 39 & 352 & 176 & 660 \\ 1 & -1 & 3 & 12 & 24 & 10 & 8 & 6 & 57 & 27 & 132 \\ 1 & 3 & -1 & 8 & 24 & 6 & 12 & 10 & 63 & 33 & 132 \\ 0 & 1 & 0 & -1 & 3 & 0 & 2 & 1 & 10 & 6 & 18 \\ 0 & 1 & 1 & 1 & -2 & 1 & 1 & 1 & 16 & 8 & 18 \\ 0 & 1 & 0 & -1 & 3 & -2 & 1 & 1 & 0 & 0 & 9 \\ 0 & 0 & 1 & 2 & 3 & 1 & -1 & 0 & 6 & 2 & 18 \\ 0 & 0 & 1 & 1 & 3 & 1 & -1 & -2 & 0 & 0 & 9 \\ 0 & 0 & 0 & 1 & \frac{15}{2} & \frac{5}{4} & 5 & \frac{17}{4} & 8 & 6 & 24 \\ 0 & 0 & 0 & 1 & -\frac{15}{2} & \frac{1}{4} & -7 & -\frac{23}{4} & -12 & -10 & -24 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{4}{3} & 0\end{pmatrix}\notag\end{equation}$$