Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-160-39-b

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This is a Fano fourfold of K3 type, appearing as case K3-30 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ1P1 × Gr(2, 4)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1 + 2w3)
Bundle
O_1(1) box O_2(2)
ZL62 label
131.x311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 39
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 160
$\chi(X, \mathrm{T}_X)$ = -21

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 160
$\mathrm{c}_1^2\mathrm{c}_2$ = 136
$\mathrm{c}_2^2$ = 114
$\mathrm{c}_1\mathrm{c}_3$ = 24
$\mathrm{c}_4$ = 30
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 22 1 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
$\mathrm{HH}^{ 2 }$ = x0 + 21 0 21 x0
$\mathrm{HH}^{ 3 }$ = x0 + 32 0 0 x0 + 1 31
$\mathrm{HH}^{ 4 }$ = 40 0 0 1 0 39
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 34t + 90t^{2} + 34t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(425)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(488)

Quantum differential equation

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

$$\begin{equation} L_X=\left( t + 1 \right) ^{3} \left( 32t^{2} - 16t + 1 \right) \left( 96t^{2} + 16t + 1 \right) \left( 3022848t^{6} + 7004160t^{5} + 3878400t^{4} - 232592t^{3} - 34260t^{2} - 408t - 5 \right)D^{6} + \left( t + 1 \right) ^{2} \left( 157865213952t^{11} + 421351391232t^{10} + 303749922816t^{9} + 455147520t^{8} - 36442329088t^{7} + 327299072t^{6} + 380461056t^{5} - 2286144t^{4} + 1313580t^{3} + 138432t^{2} + 1189t + 10 \right)D^{5} + \left( t + 1 \right) \left( 1067911741440t^{12} + 3667440697344t^{11} + 4216779177984t^{10} + 1616510877696t^{9} - 87715078144t^{8} - 91227552768t^{7} + 3744114944t^{6} + 1234455120t^{5} + 84190924t^{4} - 249696t^{3} - 133985t^{2} - 1304t - 5 \right)D^{4} + t \left( 3668044677120t^{12} + 15641188761600t^{11} + 24713677504512t^{10} + 17039638265856t^{9} + 4088727621632t^{8} - 350823575552t^{7} - 135480902400t^{6} + 4374819136t^{5} + 1745225444t^{4} + 55190112t^{3} + 1276775t^{2} + 13012t + 3 \right)D^{3} + 2 t ^{2} \left( 3361600438272t^{11} + 13935210135552t^{10} + 21117154885632t^{9} + 13625141207040t^{8} + 2883271751680t^{7} - 277995781632t^{6} - 73598029312t^{5} - 906639840t^{4} + 761173052t^{3} + 24520712t^{2} + 642425t + 5030 \right)D^{2} + 32 t ^{2} \left( 193849196544t^{11} + 788103954432t^{10} + 1157741936640t^{9} + 706323376128t^{8} + 130766092288t^{7} - 16025754368t^{6} - 3239064192t^{5} - 132285984t^{4} + 23576904t^{3} + 841068t^{2} + 21338t + 125 \right)D + 128 t ^{2} \left( 17411604480t^{11} + 69887066112t^{10} + 100466638848t^{9} + 58716301824t^{8} + 9622985088t^{7} - 1464436896t^{6} - 260370384t^{5} - 12043064t^{4} + 1289972t^{3} + 52482t^{2} + 1189t + 5 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 4)

|λ| = 13.65685 λ ≈ 13.65685; multiplicity 1; Jordan blocks 1 λ ≈ -8 - 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ -8 + 5.656854i; multiplicity 1; Jordan blocks 1 λ ≈ 2.343146; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 1, 2 λ ≈ 0; multiplicity 2; Jordan blocks 1, 1 3 2
Sector dimension
9
Spectral radius
$13.6568542$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.65685$11
$-8 - 5.656854\mathrm{i}$11
$-8 + 5.656854\mathrm{i}$11
$2.343146$11
$-1$31, 2
$0$21, 1

Characteristic polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 1\right)^{3} \left(\lambda^{2} + 16 \lambda + 96\right) \left(\lambda^{2} - 16 \lambda + 32\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 1\right)^{2} \left(\lambda^{2} + 16 \lambda + 96\right) \left(\lambda^{2} - 16 \lambda + 32\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 8 & 24 & 12 & 12 & 8 & 20 & 80 \\ 1 & -1 & 0 & 8 & 6 & 6 & 6 & 3 & 16 \\ 2 & 2 & 0 & 0 & 8 & 8 & 24 & 18 & 8 \\ 0 & 2 & 1 & -1 & 0 & 0 & 8 & 8 & 6 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 & 4 & 12 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 & 4 & 12 \\ 0 & 0 & 0 & 4 & 1 & 1 & -1 & 0 & 8 \\ 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0\end{pmatrix}\notag\end{equation}$$