Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-216-50-a

·

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

Defining data
Ambient
ℙ3P3 × Gr(2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(2w5) + E^(w3 + w4)
Bundle
O_2(2) + Q_1 box U_2^v
ZL62 label
3051.CwD
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 50
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 216
$\chi(X, \mathrm{T}_X)$ = -11

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 216
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 40
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 10 0 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 + 11 0 x0 + 11 x2
$\mathrm{HH}^{ 3 }$ = x1 + 47 0 0 x1 47
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 51 0 0 x1 - x2 + 1 0 50
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 45t + 124t^{2} + 45t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(98)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(145)

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) ^{2} \left( 12t^{3} + 68t^{2} - 2t + 1 \right) \left( 212t^{3} + 76t^{2} + 2t - 1 \right) \left( 582933857656t^{9} + 677088440208t^{8} + 338745255320t^{7} + 99214208700t^{6} + 17691571968t^{5} + 1598297376t^{4} + 80632134t^{3} + 4038120t^{2} - 3321t + 243 \right)D^{6} + \left( 2t + 1 \right) \left( 62285316822828288t^{16} + 398866505968661632t^{15} + 590061360656571392t^{14} + 432354632130180864t^{13} + 194680688814916608t^{12} + 58709281391865504t^{11} + 11976624561040368t^{10} + 1601485842442920t^{9} + 142163795748432t^{8} + 10563982318344t^{7} + 999984517728t^{6} + 116367591624t^{5} + 9118497924t^{4} + 371024514t^{3} + 16197678t^{2} - 10935t + 486 \right)D^{5} + \left( 1038088613713804800t^{17} + 6101479069688650240t^{16} + 10119512047394392992t^{15} + 8583118583819899968t^{14} + 4530216799454961760t^{13} + 1625627881060885872t^{12} + 407897979129362272t^{11} + 70598778270354528t^{10} + 8201719267952888t^{9} + 633726214186176t^{8} + 30234429251452t^{7} - 120990118932t^{6} - 189386816592t^{5} - 12997278696t^{4} - 376684686t^{3} - 14107716t^{2} + 12231t - 243 \right)D^{4} + t \left( 4359972177597980160t^{16} + 22110588981760709120t^{15} + 33628128878424550080t^{14} + 26639953611179022976t^{13} + 13299991714782431936t^{12} + 4546221597007370784t^{11} + 1084363572707348672t^{10} + 177269942364538368t^{9} + 19651183542464088t^{8} + 1545279720769680t^{7} + 91649444980416t^{6} + 3487994134128t^{5} + 43724541000t^{4} - 70226640t^{3} - 12913038t^{2} + 88776t + 243 \right)D^{3} + 4 t ^{2} \left( 2408365583816027136t^{15} + 10727698908519337216t^{14} + 15183029377035918232t^{13} + 11383010574410780464t^{12} + 5435412284720718232t^{11} + 1784461707372097044t^{10} + 406879367220971560t^{9} + 63130885318084200t^{8} + 6690543336479304t^{7} + 515734536226152t^{6} + 29956257725995t^{5} + 1001500411800t^{4} + 9296723736t^{3} + 71691774t^{2} + 1531845t + 17604 \right)D^{2} + 8 t ^{2} \left( 1307991653279394048t^{15} + 5234917502680408576t^{14} + 6994447564080343688t^{13} + 5022794482932435792t^{12} + 2316812891051965040t^{11} + 736462507233199980t^{10} + 161666339509466360t^{9} + 24006035378483652t^{8} + 2457722569022688t^{7} + 186912266441748t^{6} + 10513945597233t^{5} + 297926491122t^{4} + 1268637525t^{3} + 20692746t^{2} + 658125t + 4374 \right)D + 48 t ^{2} \left( 88979024032611840t^{15} + 328939787512801280t^{14} + 421059932369408656t^{13} + 292917954985420896t^{12} + 131707767061871944t^{11} + 40859832613712472t^{10} + 8705633612439064t^{9} + 1248953614326180t^{8} + 124688064282984t^{7} + 9403575334512t^{6} + 511990189818t^{5} + 12481580664t^{4} + 2259171t^{3} + 610938t^{2} + 15876t + 162 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 6)

|λ| = 10.8275 λ ≈ 10.8275; multiplicity 1; Jordan blocks 1 λ ≈ 1.087743 - 8.197474i; multiplicity 1; Jordan blocks 1 λ ≈ 1.087743 + 8.197474i; multiplicity 1; Jordan blocks 1 λ ≈ -4.41375 - 0.3139739i; multiplicity 1; Jordan blocks 1 λ ≈ -4.41375 + 0.3139739i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ -0.1754854; multiplicity 1; Jordan blocks 1 2
Sector dimension
8
Spectral radius
$10.8275008$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.8275$11
$1.087743 - 8.197474\mathrm{i}$11
$1.087743 + 8.197474\mathrm{i}$11
$-4.41375 - 0.3139739\mathrm{i}$11
$-4.41375 + 0.3139739\mathrm{i}$11
$-2$22
$-0.1754854$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{3} - 2 \lambda^{2} + 68 \lambda + 12\right) \left(\lambda^{3} - 2 \lambda^{2} - 76 \lambda - 212\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{3} - 2 \lambda^{2} + 68 \lambda + 12\right) \left(\lambda^{3} - 2 \lambda^{2} - 76 \lambda - 212\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 6 & 6 & 32 & 64 & 80 \\ 2 & 0 & 4 & 4 & 0 & -6 & -12 & 0 \\ 1 & 0 & 0 & 0 & 0 & 6 & 18 & 32 \\ 0 & 1 & 4 & 0 & 0 & 0 & 0 & 6 \\ 0 & 2 & -1 & 4 & 0 & 4 & 4 & -6 \\ 0 & 0 & 0 & 1 & -2 & -4 & -4 & -4 \\ 0 & 0 & 0 & 3 & 3 & 4 & 4 & 4 \\ 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$