Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-64-20-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(2w3) + E^(w1 + w2 + 2w3)
Bundle
O_3(2) + O_1(1) box O_2(1) box O_3(2)
ZL62 label
1140.8x4001
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 20
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 64
$\chi(X, \mathrm{T}_X)$ = -39

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 64
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 152
$\mathrm{c}_1\mathrm{c}_3$ = 0
$\mathrm{c}_4$ = 72
Hodge diamond
1
00
0 3 0
0 0 0 0
0 5 54 5 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 39 0 x0 + 39 x2
$\mathrm{HH}^{ 3 }$ = x1 0 0 x1 0
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 32 0 0 x1 - x2 + 11 1 20
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 15t + 32t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(692)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(689)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t + 1 \right) \left( 10t - 1 \right) \left( 6t + 1 \right) ^{4} \left( 244t^{2} + 32t - 1 \right) \left( 579387744t^{6} + 359936592t^{5} + 76159920t^{4} + 6051976t^{3} + 75678t^{2} - 5943t + 8 \right)D^{6} + \left( 6t + 1 \right) ^{3} \left( 356253936030720t^{11} + 407656901938176t^{10} + 173502315641088t^{9} + 33645214427136t^{8} + 2635633273984t^{7} - 20983360128t^{6} - 11699539680t^{5} - 293018112t^{4} - 5767004t^{3} - 450756t^{2} + 17717t - 16 \right)D^{5} + \left( 6t + 1 \right) ^{2} \left( 17812696801536000t^{12} + 20861229183390720t^{11} + 9622795169051136t^{10} + 2217321412826880t^{9} + 261739036782336t^{8} + 13105264231808t^{7} - 90497824960t^{6} - 23854043552t^{5} + 442827496t^{4} + 65937140t^{3} + 519434t^{2} - 10821t + 8 \right)D^{4} + t \left( 6t + 1 \right) \left( 448879959398707200t^{12} + 551206407455170560t^{11} + 278109315088680960t^{10} + 74297732773865472t^{9} + 11212410919940352t^{8} + 919882253832192t^{7} + 31560812060416t^{6} - 432967958272t^{5} - 46219665648t^{4} + 100752032t^{3} + 21374152t^{2} - 312984t - 57 \right)D^{3} + 8 t ^{2} \left( 743858218432143360t^{12} + 974886228667551744t^{11} + 541138996686223872t^{10} + 165460124618164224t^{9} + 30243964168820736t^{8} + 3321233107442880t^{7} + 201899521072512t^{6} + 4524827545744t^{5} - 139880287168t^{4} - 7544980572t^{3} - 8075242t^{2} + 396655t - 21120 \right)D^{2} + 32 t ^{2} \left( 201995981729418240t^{12} + 252291045503273472t^{11} + 133231077991786752t^{10} + 38663548141848192t^{9} + 6680215637048736t^{8} + 687890465007408t^{7} + 38401101369408t^{6} + 691346620928t^{5} - 32015115010t^{4} - 1399881151t^{3} - 3988766t^{2} - 111374t - 1460 \right)D + 432 t ^{2} \left( 6107210331955200t^{12} + 7373724863016960t^{11} + 3755514183985152t^{10} + 1048227857164800t^{9} + 173449789931136t^{8} + 16962173266368t^{7} + 878476975776t^{6} + 12086648240t^{5} - 872412968t^{4} - 32841788t^{3} - 145130t^{2} - 4105t - 16 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ4P4

|λ| = 38.36068 λ ≈ 38.36068; multiplicity 1; Jordan blocks 1 λ ≈ 10; multiplicity 2; Jordan blocks 1, 1 λ ≈ -6.36068; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 7; Jordan blocks 1, 1, 2, 3 λ ≈ -2; multiplicity 1; Jordan blocks 1 2 7
Sector dimension
12
Spectral radius
$38.3606798$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$38.36068$11
$10$21, 1
$-6.36068$11
$-6$71, 1, 2, 3
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 6\right)^{7} \left(\lambda - 10\right)^{2} \left(\lambda^{2} - 32 \lambda - 244\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 6\right)^{3} \left(\lambda - 10\right) \left(\lambda^{2} - 32 \lambda - 244\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 18 & 18 & 72 & 72 & 324 & 324 & 720 & 1728 & 3456 & 3456 & 17280 \\ 1 & -1 & 1 & 6 & 16 & 64 & 24 & 76 & 288 & 576 & 384 & 2592 \\ 1 & 1 & -1 & 6 & 16 & 24 & 64 & 76 & 288 & 384 & 576 & 2592 \\ 1 & 2 & 2 & 4 & 4 & 34 & 34 & 88 & 144 & 360 & 360 & 1728 \\ 0 & 1 & 1 & 0 & -2 & 6 & 6 & 4 & 56 & 68 & 68 & 432 \\ 0 & 1 & 0 & 1 & 2 & 3 & 1 & 8 & 32 & 72 & 32 & 288 \\ 0 & 0 & 1 & 1 & 2 & 1 & 3 & 8 & 32 & 32 & 72 & 288 \\ 0 & 0 & 0 & 1 & 0 & 2 & 2 & 0 & 4 & 18 & 18 & 72 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & -\frac{1}{2} & 2 & 7 & 7 & 54 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & \frac{3}{2} & 2 & 0 & 2 & 18 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 2 & 2 & 0 & 18 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$