Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-40-15-b

·

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

Defining data
Ambient
ℙ1P1 × ℙ7P7
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
3 copies of E^(2w2) + E^(w1 + w2)
Bundle
3 copies of O_2(2) + O_1(1) box O_2(1)
ZL62 label
1700.999w0
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;8) \\ \det\mathcal{E} &\cong \mathcal{O}(1;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 40
$\chi(X, \mathrm{T}_X)$ = -48

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 40
$\mathrm{c}_1^2\mathrm{c}_2$ = 88
$\mathrm{c}_2^2$ = 168
$\mathrm{c}_1\mathrm{c}_3$ = -24
$\mathrm{c}_4$ = 72
Hodge diamond
1
00
0 2 0
0 0 0 0
0 4 58 4 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 + 48 0 x0 + 48 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 27 0 0 x1 - x2 + 12 0 15
$\mathrm{HH}^{ 5 }$ = 3 0 0 3 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 18t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(732)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(724)

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( 40t - 1 \right) \left( 72t - 1 \right) \left( 8t + 1 \right) ^{3} \left( 9t + 1 \right) ^{3} \left( 69972553728t^{6} + 26529831936t^{5} + 3154222464t^{4} + 98438800t^{3} - 3681756t^{2} - 130680t + 125 \right)D^{6} + \left( 8t + 1 \right) ^{2} \left( 9t + 1 \right) ^{2} \left( 304699683561799680t^{10} + 147103782448398336t^{9} + 25226529021591552t^{8} + 1686350604570624t^{7} + 10626248887296t^{6} - 2721613543552t^{5} - 47499370464t^{4} - 27987644t^{3} + 12274504t^{2} + 381915t - 250 \right)D^{5} + \left( 8t + 1 \right) \left( 9t + 1 \right) \left( 182819810137079808000t^{12} + 117740612593165271040t^{11} + 30265066929494163456t^{10} + 3888247728985473024t^{9} + 243750887502815232t^{8} + 4115413737412608t^{7} - 301551685398016t^{6} - 14875974718096t^{5} - 151391760956t^{4} + 1791491312t^{3} - 8593895t^{2} - 240760t + 125 \right)D^{4} + t \left( 55284710585452933939200t^{13} + 45405958669562900643840t^{12} + 15804212559476875591680t^{11} + 3015278366818975875072t^{10} + 336916714829240696832t^{9} + 21098493715738411008t^{8} + 518241819639054336t^{7} - 18028902518017408t^{6} - 1617448086967008t^{5} - 41043534028052t^{4} - 344584304792t^{3} + 526149513t^{2} - 13669708t - 1875 \right)D^{3} + 2 t ^{2} \left( 61076442170595622256640t^{12} + 47413943285061846564864t^{11} + 15455236822199889297408t^{10} + 2728173621422525644800t^{9} + 276321262167709876224t^{8} + 14898711463638700032t^{7} + 216208885169533440t^{6} - 20448578589225664t^{5} - 1222058881304608t^{4} - 25676768752620t^{3} - 177635675072t^{2} + 176119015t - 3766230 \right)D^{2} + 64 t ^{2} \left( 2073176646954485022720t^{12} + 1541040501551901179904t^{11} + 476943943968771342336t^{10} + 79037536355373846528t^{9} + 7365104255666483712t^{8} + 344958393921715584t^{7} + 1669797189783168t^{6} - 633017984618380t^{5} - 29802912996866t^{4} - 535256025860t^{3} - 2991559643t^{2} - 244855t - 33925 \right)D + 384 t ^{2} \left( 141032424962890137600t^{12} + 101690161455295365120t^{11} + 30315952026600210432t^{10} + 4792317461591298048t^{9} + 418269868119848448t^{8} + 17304350789572992t^{7} - 71660008703328t^{6} - 38872951656632t^{5} - 1564112394328t^{4} - 24793601734t^{3} - 114386650t^{2} - 179450t - 875 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ7P7

|λ| = 72 λ ≈ 72; multiplicity 1; Jordan blocks 1 λ ≈ 40; multiplicity 1; Jordan blocks 1 λ ≈ -9; multiplicity 3; Jordan blocks 1, 2 λ ≈ -8; multiplicity 3; Jordan blocks 3 3 3
Sector dimension
8
Spectral radius
$72$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$72$11
$40$11
$-9$31, 2
$-8$33

Characteristic polynomial

$$\begin{equation}\left(\lambda + 8\right)^{3} \left(\lambda + 9\right)^{3} \left(\lambda - 40\right) \left(\lambda - 72\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 8\right)^{3} \left(\lambda + 9\right)^{2} \left(\lambda - 40\right) \left(\lambda - 72\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 320 & 912 & 11232 & 40384 & 172224 & 623360 \\ 1 & -1 & 8 & 304 & 528 & 10320 & 15696 & 131840 \\ 1 & 1 & 16 & 48 & 848 & 2400 & 12720 & 40384 \\ 0 & 1 & 1 & 15 & 24 & 800 & 1024 & 10320 \\ 0 & 0 & 1 & 1 & 16 & 48 & 352 & 912 \\ 0 & 0 & 0 & 1 & 1 & 15 & 24 & 304 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 16 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$