Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-52-18-a

·

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

Defining data
Ambient
ℙ3P3 × ℙ5P5
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_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w3 + w4) + E^(2w1 + 2w4)
Bundle
Q_1 box O_2(1) + O_1(2) box O_2(2)
ZL62 label
3050.wDx328
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(3;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 52
$\chi(X, \mathrm{T}_X)$ = -77

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 52
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 196
$\mathrm{c}_1\mathrm{c}_3$ = -48
$\mathrm{c}_4$ = 168
Hodge diamond
1
00
0 2 0
0 0 0 0
0 18 126 18 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 + 77 0 77 x0
$\mathrm{HH}^{ 3 }$ = x0 + 6 0 0 x0 6
$\mathrm{HH}^{ 4 }$ = 95 0 0 77 0 18
$\mathrm{HH}^{ 5 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 6 }$ = 1 0 1 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 24t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(687)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(685)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t + 1 \right) ^{5} \left( 100t^{2} + 40t - 1 \right) \left( 508t^{2} + 40t + 1 \right) \left( 137270000t^{4} - 59026664t^{3} + 15232548t^{2} + 166618t + 221 \right)D^{6} + 2 \left( 2t + 1 \right) ^{4} \left( 146439636000000t^{9} + 25651289113600t^{8} - 12391498151040t^{7} + 6038916307328t^{6} + 1784295131248t^{5} + 58684233552t^{4} + 273679088t^{3} + 29922448t^{2} + 247938t + 221 \right)D^{5} + \left( 2t + 1 \right) ^{3} \left( 4881321200000000t^{10} + 1836892420288000t^{9} - 369884793666560t^{8} + 128823193997824t^{7} + 122294920508096t^{6} + 13202554588192t^{5} + 263701752208t^{4} + 2824252472t^{3} - 55076864t^{2} - 391954t - 221 \right)D^{4} + 8 t \left( 2t + 1 \right) ^{2} \left( 5125387260000000t^{10} + 3287911472416000t^{9} - 35540491780800t^{8} + 10921532355680t^{7} + 192508868187904t^{6} + 48676411077960t^{5} + 2957906211104t^{4} + 45250845760t^{3} + 314932200t^{2} + 531575t + 102 \right)D^{3} + 8 \left( 2t + 1 \right) t ^{2} \left( 22649330368000000t^{10} + 21785061446822400t^{9} + 3767751798236160t^{8} - 301995180673024t^{7} + 974838788996768t^{6} + 513811459265328t^{5} + 72340866486416t^{4} + 2905362453764t^{3} + 38861648310t^{2} + 234060283t + 461747 \right)D^{2} + 16 t ^{2} \left( 24601858848000000t^{11} + 32725075377228800t^{10} + 12066649479790080t^{9} + 620611376354624t^{8} + 957342862706560t^{7} + 1006416052927600t^{6} + 275046765169112t^{5} + 26239670196664t^{4} + 771082564420t^{3} + 9563632514t^{2} + 45095321t + 88621 \right)D + 32 t ^{2} \left( 5020787520000000t^{11} + 6232011969792000t^{10} + 1966075794109440t^{9} + 10589680455744t^{8} + 209464958389696t^{7} + 192316304916192t^{6} + 45432442594456t^{5} + 3604660503212t^{4} + 85975736362t^{3} + 993884750t^{2} + 3506199t + 5525 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ5P5

|λ| = 42.36068 λ ≈ 42.36068; multiplicity 1; Jordan blocks 1 λ ≈ -20 - 10.3923i; multiplicity 1; Jordan blocks 1 λ ≈ -20 + 10.3923i; multiplicity 1; Jordan blocks 1 λ ≈ -2.36068; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 5; Jordan blocks 1, 1, 3 5
Sector dimension
9
Spectral radius
$42.3606798$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$42.36068$11
$-20 - 10.3923\mathrm{i}$11
$-20 + 10.3923\mathrm{i}$11
$-2.36068$11
$-2$51, 1, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{5} \left(\lambda^{2} + 40 \lambda + 508\right) \left(\lambda^{2} - 40 \lambda - 100\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{2} + 40 \lambda + 508\right) \left(\lambda^{2} - 40 \lambda - 100\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 48 & 52 & 792 & 384 & 1008 & 9152 & 7296 & 42880 \\ 1 & -2 & 8 & 68 & 84 & 56 & 1728 & 1200 & 3712 \\ 1 & 2 & 0 & 144 & 92 & 160 & 1152 & 816 & 5440 \\ 0 & 1 & 1 & -2 & -4 & 4 & 64 & 68 & 384 \\ 0 & 1 & 0 & 8 & -2 & 12 & 8 & 20 & 432 \\ 0 & 0 & 1 & 2 & 6 & -2 & 144 & 100 & 192 \\ 0 & 0 & 0 & 1 & -\frac{1}{2} & 2 & -8 & -2 & 44 \\ 0 & 0 & 0 & 1 & 2 & 0 & 16 & 6 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$