Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-256-57-a

·

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

Varieties in this family can be blown up to

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ3P3
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_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(w3 + 2w5) + E^(w1 + w4)
Bundle
O_2(1) box O_3(2) + O_1(1) box Q_2
ZL62 label
202030.x3C1w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 2
$\mathrm{h}^0(X, \omega_X^\vee)$ = 57
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 256
$\chi(X, \mathrm{T}_X)$ = -6

$\omega_X^\vee \cong \mathcal{O}_X(2)$

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 256
$\mathrm{c}_1^2\mathrm{c}_2$ = 160
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 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 + x1 + 6 0 x0 + 6 x1
$\mathrm{HH}^{ 3 }$ = x1 + 48 0 0 x1 - 18 66
$\mathrm{HH}^{ 4 }$ = 57 0 0 0 0 57
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 7t + 7t^{2} + t^{3} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(259)
Coates et al. (2020) MW(5)
Coates–Kasprzyk–Prince (2015) CKP(327)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 32t^{4} - 144t^{3} + 40t^{2} - 12t + 1 \right) \left( 32t^{4} + 144t^{3} + 40t^{2} + 12t + 1 \right) \left( 1379024896t^{8} - 181690112t^{6} + 32203856t^{4} + 160775t^{2} + 136 \right)D^{6} + 2 \left( 14121214935040t^{16} - 152437496479744t^{14} + 18570092085248t^{12} - 3527276827648t^{10} - 277037824256t^{8} - 556812800t^{6} - 140567296t^{4} - 499733t^{2} - 272 \right)D^{5} + 4 \left( 54366677499904t^{16} - 340827602288640t^{14} + 64054093561856t^{12} - 13315483720192t^{10} - 406200526464t^{8} - 986296160t^{6} + 122964452t^{4} + 370629t^{2} + 136 \right)D^{4} + 8 t ^{2} \left( 103084869025792t^{14} - 369261049675776t^{12} + 91720177627136t^{10} - 21127821675520t^{8} - 305989253824t^{6} - 1259677440t^{4} - 1614768t^{2} - 5831 \right)D^{3} + 128 t ^{2} \left( 12720125640704t^{14} - 26591028115456t^{12} + 8287153333632t^{10} - 2016286904136t^{8} - 15422020074t^{6} - 337096385t^{4} - 533452t^{2} - 102 \right)D^{2} + 256 t ^{4} \left( 6200095932416t^{12} - 7989995554816t^{10} + 3020985441920t^{8} - 739274260600t^{6} - 4109085596t^{4} - 179641501t^{2} - 331640 \right)D + 1536 t ^{4} \left( 386126970880t^{12} - 333456404480t^{10} + 147420386560t^{8} - 34919816144t^{6} - 204213358t^{4} - 10473275t^{2} - 20944 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ3P3

|λ| = 9.327276 λ ≈ -9.327276; multiplicity 1; Jordan blocks 1 λ ≈ 9.327276; multiplicity 1; Jordan blocks 1 λ ≈ -1.218011 - 3.607022i; multiplicity 1; Jordan blocks 1 λ ≈ -1.218011 + 3.607022i; multiplicity 1; Jordan blocks 1 λ ≈ 1.218011 - 3.607022i; multiplicity 1; Jordan blocks 1 λ ≈ 1.218011 + 3.607022i; multiplicity 1; Jordan blocks 1 λ ≈ -0.2367022; multiplicity 1; Jordan blocks 1 λ ≈ 0.2367022; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$9.32727615$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-9.327276$11
$9.327276$11
$-1.218011 - 3.607022\mathrm{i}$11
$-1.218011 + 3.607022\mathrm{i}$11
$1.218011 - 3.607022\mathrm{i}$11
$1.218011 + 3.607022\mathrm{i}$11
$-0.2367022$11
$0.2367022$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda^{4} + 12 \lambda^{3} + 40 \lambda^{2} + 144 \lambda + 32\right) \left(\lambda^{4} - 12 \lambda^{3} + 40 \lambda^{2} - 144 \lambda + 32\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda^{4} + 12 \lambda^{3} + 40 \lambda^{2} + 144 \lambda + 32\right) \left(\lambda^{4} - 12 \lambda^{3} + 40 \lambda^{2} - 144 \lambda + 32\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 4 & 0 & 0 & 0 & 16 & 16 & 0 \\ 2 & 0 & 0 & 0 & 4 & 6 & 0 & 0 & 8 \\ 2 & 0 & 0 & 4 & 2 & 4 & 0 & 0 & 16 \\ 0 & 2 & 0 & 0 & 0 & 0 & 4 & 6 & 0 \\ 0 & 2 & 2 & 0 & 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 4 & 2 & 0 \\ 0 & 0 & 0 & 2 & 2 & -\frac{1}{2} & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 2 & 3 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0\end{pmatrix}\notag\end{equation}$$