Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-20-11-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ5P5
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(4w2) + E^(w1 + w2)
Bundle
O_2(4) + O_1(1) box O_2(1)
ZL62 label
150.Jg
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 11
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 20
$\chi(X, \mathrm{T}_X)$ = -98

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 20
$\mathrm{c}_1^2\mathrm{c}_2$ = 80
$\mathrm{c}_2^2$ = 240
$\mathrm{c}_1\mathrm{c}_3$ = -88
$\mathrm{c}_4$ = 212
Hodge diamond
1
00
0 2 0
0 0 0 0
0 22 162 22 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 + 98 0 x0 + 98 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 123 0 0 x1 - x2 + 112 0 11
$\mathrm{HH}^{ 5 }$ = x3 0 0 x3 0
$\mathrm{HH}^{ 6 }$ = x3 - 9 0 x3 - 9 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 6t + 6t^{2} + 6t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(744)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(733)

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( 200t - 1 \right) \left( 264t - 1 \right) \left( 24t + 1 \right) ^{3} \left( 25t + 1 \right) ^{3} \left( 420678937497600t^{6} + 40427923405824t^{5} + 419949283968t^{4} - 61219019376t^{3} - 1945637820t^{2} - 13166488t + 4205 \right)D^{6} + \left( 24t + 1 \right) ^{2} \left( 25t + 1 \right) ^{2} \left( 279869283538403328000000t^{10} + 37580334194169446400000t^{9} + 1244445936628071628800t^{8} - 38779269176804069376t^{7} - 3306394302087508992t^{6} - 69211235698978176t^{5} - 457687724493600t^{4} - 343176339676t^{3} + 9260813784t^{2} + 37342299t - 8410 \right)D^{5} + \left( 24t + 1 \right) \left( 25t + 1 \right) \left( 1399346417692016640000000000t^{12} + 263333273063532134400000000t^{11} + 16897258254529595473920000t^{10} + 198458433112422393446400t^{9} - 27498392331491562725376t^{8} - 1517235436827696642048t^{7} - 34168594825782148608t^{6} - 365563443703584528t^{5} - 1596853165009180t^{4} - 1011596135024t^{3} - 6491942519t^{2} - 24525464t + 4205 \right)D^{4} + t \left( 3526352972583881932800000000000t^{13} + 874665597283835019264000000000t^{12} + 84356541427206091898880000000t^{11} + 3394814597742686288117760000t^{10} - 23780587165284213883699200t^{9} - 8541110205006434033762304t^{8} - 397769436283309592942592t^{7} - 9474924138388332872832t^{6} - 126990742735149584160t^{5} - 907446105616466932t^{4} - 2754388684378248t^{3} - 584069239703t^{2} - 5472182972t - 291827 \right)D^{3} + 2 t ^{2} \left( 3895780426854574325760000000000t^{12} + 901281733720628546764800000000t^{11} + 78877196910549564899328000000t^{10} + 2563181493749184789135360000t^{9} - 63447763849014910950604800t^{8} - 8768584313610464927600640t^{7} - 346990036097371051676160t^{6} - 7283538727408326081600t^{5} - 85764662712826601376t^{4} - 526699322496944780t^{3} - 1272621862307696t^{2} + 259988040807t - 1497178230 \right)D^{2} + 64 t ^{2} \left( 132238236471895572480000000000t^{12} + 28975824918165255782400000000t^{11} + 2341068623666090505216000000t^{10} + 61378183557993569195520000t^{9} - 2797276957118901089164800t^{8} - 274147955547333821039232t^{7} - 9654551382802615170624t^{6} - 183165618383242735428t^{5} - 1933851234240508494t^{4} - 10385292358269520t^{3} - 20186863702297t^{2} + 4433381765t - 13298530 \right)D + 384 t ^{2} \left( 8995798399448678400000000000t^{12} + 1896817341760929792000000000t^{11} + 144353114113341007872000000t^{10} + 3103565584612573877760000t^{9} - 212645259298125811468800t^{8} - 17419859419388652224640t^{7} - 565078005968973636960t^{6} - 9921933814802100552t^{5} - 95928715766292312t^{4} - 459438859870282t^{3} - 727409560906t^{2} - 24623668t - 349015 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ5P5

|λ| = 264 λ ≈ 264; multiplicity 1; Jordan blocks 1 λ ≈ 200; multiplicity 1; Jordan blocks 1 λ ≈ -25; multiplicity 3; Jordan blocks 1, 2 λ ≈ -24; multiplicity 3; Jordan blocks 3 3 3
Sector dimension
8
Spectral radius
$264$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$264$11
$200$11
$-25$31, 2
$-24$33

Characteristic polynomial

$$\begin{equation}\left(\lambda + 24\right)^{3} \left(\lambda + 25\right)^{3} \left(\lambda - 200\right) \left(\lambda - 264\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 24\right)^{3} \left(\lambda + 25\right)^{2} \left(\lambda - 200\right) \left(\lambda - 264\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 48 & 3936 & 11664 & 520416 & 2023872 & 20748480 & 92286720 \\ 1 & -1 & 24 & 3888 & 6672 & 508752 & 709200 & 18724608 \\ 1 & 1 & 80 & 208 & 13808 & 40800 & 549552 & 2023872 \\ 0 & 1 & 1 & 79 & 104 & 13600 & 16384 & 508752 \\ 0 & 0 & 1 & 1 & 80 & 208 & 4096 & 11664 \\ 0 & 0 & 0 & 1 & 1 & 79 & 104 & 3888 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 48 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$