Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-46-16-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ3P3
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + 2w4) + E^(2w1 + w4)
Bundle
O_1(1) box O_2(2) + O_1(2) box O_2(1)
ZL62 label
3030.x321x327
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 16
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 46
$\chi(X, \mathrm{T}_X)$ = -48

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 46
$\mathrm{c}_1^2\mathrm{c}_2$ = 88
$\mathrm{c}_2^2$ = 175
$\mathrm{c}_1\mathrm{c}_3$ = -18
$\mathrm{c}_4$ = 93
Hodge diamond
1
00
0 2 0
0 0 0 0
0 8 71 8 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 48 0 48 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 44 0 0 28 0 16
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 11t + 22t^{2} + 11t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(708)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(705)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 50t - 1 \right) \left( 126t + 23 \right) ^{2} \left( 4t + 1 \right) ^{3} \left( 59t^{2} + 19t + 1 \right)D^{4} + 2 t \left( 126t + 23 \right) \left( 4t + 1 \right) ^{2} \left( 7434000t^{4} + 4168056t^{3} + 772548t^{2} + 51386t + 563 \right)D^{3} + t \left( 4t + 1 \right) \left( 26227152000t^{6} + 22791420288t^{5} + 7793958424t^{4} + 1309875884t^{3} + 108259138t^{2} + 3634937t + 16652 \right)D^{2} + t \left( 149869440000t^{7} + 155045302272t^{6} + 65394272256t^{5} + 14316902832t^{4} + 1700409104t^{3} + 102105648t^{2} + 2342228t + 3703 \right)D + 16 t ^{2} \left( 4496083200t^{6} + 4416799968t^{5} + 1756119304t^{4} + 358376196t^{3} + 38915182t^{2} + 2050611t + 35972 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ3P3

|λ| = 50 λ ≈ 50; multiplicity 1; Jordan blocks 1 λ ≈ -15.09017; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 λ ≈ -3.90983; multiplicity 2; Jordan blocks 1, 1 2 4 2
Sector dimension
9
Spectral radius
$50$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$50$11
$-15.09017$21, 1
$-4$41, 3
$-3.90983$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{4} \left(\lambda - 50\right) \left(\lambda^{2} + 19 \lambda + 59\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda - 50\right) \left(\lambda^{2} + 19 \lambda + 59\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 68 & 68 & 1320 & 936 & 936 & 17248 & 17248 & 46400 \\ 1 & 0 & 7 & 148 & 110 & 90 & 1755 & 1710 & 4928 \\ 1 & 7 & 0 & 148 & 90 & 110 & 1710 & 1755 & 4928 \\ 0 & 1 & 1 & 0 & 5 & 5 & 140 & 140 & 336 \\ 0 & 1 & 0 & 7 & -2 & 7 & 78 & 90 & 240 \\ 0 & 0 & 1 & 7 & 7 & -2 & 90 & 78 & 240 \\ 0 & 0 & 0 & 1 & -\frac{4}{21} & \frac{31}{21} & 0 & 7 & \frac{136}{7} \\ 0 & 0 & 0 & 1 & \frac{31}{21} & -\frac{4}{21} & 7 & 0 & \frac{136}{7} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{2} & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$