Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-100-27-c

·

This is a Fano fourfold of K3 type, appearing as case K3-33 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ5P5 × Gr(2, 4)
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_7 (marked)node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w1 + w6) + E^(w1 + w7) + E^(w1 + w8)
Bundle
O_1(1) box U_2^v + O_1(1) box O_2(1) + O_1(1) box Q_2
ZL62 label
3150.wDw8w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(6;4) \\ \det\mathcal{E} &\cong \mathcal{O}(5;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 100
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 100
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 20
$\mathrm{c}_4$ = 32
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 24 1 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 + 23 0 x0 + 23 x1
$\mathrm{HH}^{ 3 }$ = x1 + 20 0 0 x1 + 11 9
$\mathrm{HH}^{ 4 }$ = 28 0 0 1 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 54t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(597)
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 $L_X\widehat{\mathrm{G}}_X=0$.

$$\begin{equation} L_X=\left( t + 1 \right) \left( 10t + 1 \right) \left( 2t + 1 \right) ^{2} \left( 46t + 25 \right) ^{2} \left( 28t^{2} + 20t - 1 \right)D^{4} + 2 t \left( 2t + 1 \right) \left( 46t + 25 \right) \left( 128800t^{5} + 309584t^{4} + 267424t^{3} + 98304t^{2} + 13158t + 171 \right)D^{3} + t \left( 82947200t^{7} + 259017536t^{6} + 328836160t^{5} + 216488960t^{4} + 77423520t^{3} + 14103800t^{2} + 1014434t + 5525 \right)D^{2} + 2 t \left( 59248000t^{7} + 171942848t^{6} + 201874416t^{5} + 121751416t^{4} + 39197500t^{3} + 6183750t^{2} + 343850t + 625 \right)D + 24 t ^{2} \left( 2369920t^{6} + 6554816t^{5} + 7297608t^{4} + 4136820t^{3} + 1232398t^{2} + 173475t + 7500 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ5P5 × Gr(2, 4)

|λ| = 21.31371 λ ≈ 21.31371; multiplicity 1; Jordan blocks 1 λ ≈ -10; multiplicity 2; Jordan blocks 1, 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 λ ≈ -1.313708; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 1, 1 2 3 3
Sector dimension
10
Spectral radius
$21.3137085$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$21.31371$11
$-10$21, 1
$-2$31, 2
$-1.313708$11
$-1$31, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda + 2\right)^{3} \left(\lambda + 10\right)^{2} \left(\lambda^{2} - 20 \lambda - 28\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right) \left(\lambda + 2\right)^{2} \left(\lambda + 10\right) \left(\lambda^{2} - 20 \lambda - 28\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 18 & 18 & 120 & 96 & 48 & 48 & 392 & 784 & 840 \\ 1 & -1 & 3 & 28 & 28 & 8 & 8 & 66 & 120 & 168 \\ 1 & 3 & -1 & 28 & 16 & 14 & 14 & 60 & 132 & 168 \\ 0 & 1 & 1 & -2 & 0 & 0 & 0 & 14 & 28 & 18 \\ 0 & 1 & 0 & 3 & -1 & 2 & 2 & 6 & 16 & 18 \\ 0 & 0 & 1 & 3 & 4 & -1 & 0 & 8 & 12 & 18 \\ 0 & 0 & 1 & 3 & 4 & 0 & -1 & 8 & 12 & 18 \\ 0 & 0 & 0 & 2 & -\frac{10}{7} & \frac{16}{7} & \frac{16}{7} & -1 & 6 & \frac{54}{7} \\ 0 & 0 & 0 & 1 & \frac{16}{7} & -\frac{5}{14} & -\frac{5}{14} & \frac{3}{2} & -1 & \frac{27}{7} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & \frac{14}{3} & 0\end{pmatrix}\notag\end{equation}$$