Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-40-15-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ4P4
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 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^(2w1 + 2w3)
Bundle
O_2(2) + O_1(2) box O_2(2)
ZL62 label
2040.8x318
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 40
$\chi(X, \mathrm{T}_X)$ = -65

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 40
$\mathrm{c}_1^2\mathrm{c}_2$ = 88
$\mathrm{c}_2^2$ = 196
$\mathrm{c}_1\mathrm{c}_3$ = -36
$\mathrm{c}_4$ = 144
Hodge diamond
1
00
0 2 0
0 0 0 0
0 15 108 15 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 + 65 0 x0 + 65 0
$\mathrm{HH}^{ 3 }$ = 3 0 0 3 0
$\mathrm{HH}^{ 4 }$ = 80 0 0 65 0 15
$\mathrm{HH}^{ 5 }$ = 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

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

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(724)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(717)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t + 1 \right) \left( 6t + 1 \right) ^{5} \left( 13112t^{3} + 1140t^{2} + 42t - 1 \right) \left( 80385812t^{4} + 7197608t^{3} + 769080t^{2} + 2264t + 5 \right)D^{6} + 2 \left( 6t + 1 \right) ^{4} \left( 132806364634944t^{9} + 86099911767616t^{8} + 15961355640224t^{7} + 1743005649920t^{6} + 109628957732t^{5} + 3161136316t^{4} + 12728260t^{3} + 1555264t^{2} + 3461t + 5 \right)D^{5} + \left( 6t + 1 \right) ^{3} \left( 13280636463494400t^{10} + 8753387608187520t^{9} + 1950985921290560t^{8} + 241863730237952t^{7} + 19094257553872t^{6} + 777328219528t^{5} + 9302271220t^{4} - 4547428t^{3} - 2834092t^{2} - 5992t - 5 \right)D^{4} + 2 t \left( 6t + 1 \right) ^{2} \left( 167336019440029440t^{10} + 117688594942851840t^{9} + 31418518535083200t^{8} + 4618275046756096t^{7} + 438496809506448t^{6} + 25058879803536t^{5} + 654589667540t^{4} + 5045885376t^{3} + 26414724t^{2} + 107656t + 27 \right)D^{3} + 4 \left( 6t + 1 \right) t ^{2} \left( 1109198757431052288t^{10} + 861042773903810688t^{9} + 271935574395641856t^{8} + 47933686450198560t^{7} + 5442402746815904t^{6} + 404035626969208t^{5} + 17018681584532t^{4} + 306529593818t^{3} + 1826543872t^{2} + 17602193t + 45390 \right)D^{2} + 16 t ^{2} \left( 1807229009952317952t^{11} + 1580017903071474816t^{10} + 584882699830508736t^{9} + 123076679832783936t^{8} + 16729754354732928t^{7} + 1539376434584624t^{6} + 91122954562442t^{5} + 2951091535762t^{4} + 39515609613t^{3} + 226172288t^{2} + 2052700t + 4195 \right)D + 96 t ^{2} \left( 122940748976348160t^{11} + 101781031456846080t^{10} + 35780697614079360t^{9} + 7181046909361440t^{8} + 935498407232496t^{7} + 82245903131768t^{6} + 4542364643942t^{5} + 130235071330t^{4} + 1438760925t^{3} + 8824394t^{2} + 55339t + 85 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 63.28692 λ ≈ 63.28692; multiplicity 1; Jordan blocks 1 λ ≈ -10.64346 - 9.690212i; multiplicity 1; Jordan blocks 1 λ ≈ -10.64346 + 9.690212i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 5; Jordan blocks 1, 1, 3 λ ≈ -2; multiplicity 1; Jordan blocks 1 5
Sector dimension
9
Spectral radius
$63.2869178$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$63.28692$11
$-10.64346 - 9.690212\mathrm{i}$11
$-10.64346 + 9.690212\mathrm{i}$11
$-6$51, 1, 3
$-2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 6\right)^{5} \left(\lambda^{3} - 42 \lambda^{2} - 1140 \lambda - 13112\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 6\right)^{3} \left(\lambda^{3} - 42 \lambda^{2} - 1140 \lambda - 13112\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 84 & 120 & 2592 & 1008 & 2592 & 52416 & 27072 & 305280 \\ 1 & 0 & 12 & 184 & 80 & 184 & 4272 & 2256 & 25344 \\ 1 & 6 & 4 & 228 & 100 & 200 & 4656 & 2400 & 27072 \\ 0 & 1 & 1 & 4 & 2 & 16 & 232 & 128 & 1584 \\ 0 & 1 & 0 & 12 & -2 & 8 & 152 & 72 & 1008 \\ 0 & 0 & 1 & 6 & 4 & 0 & 180 & 100 & 1008 \\ 0 & 0 & 0 & 1 & 0 & 2 & 6 & 6 & 84 \\ 0 & 0 & 0 & 1 & 1 & -1 & 10 & -2 & 36 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0\end{pmatrix}\notag\end{equation}$$