Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-432-90-b

·

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

Defining data
Ambient
ℙ4P4 × ℙ1P1
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)ω1ω2ω3ω4ω5
Bundle
E^(2w1)
Bundle
O_1(2)
ZL62 label
140.6
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;2) \\ \det\mathcal{E} &\cong \mathcal{O}(2;0) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 90
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 432
$\chi(X, \mathrm{T}_X)$ = 13

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 432
$\mathrm{c}_1^2\mathrm{c}_2$ = 204
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 56
$\mathrm{c}_4$ = 8
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 2 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 13 0 13
$\mathrm{HH}^{ 2 }$ = 65 0 0 65
$\mathrm{HH}^{ 3 }$ = 135 0 0 0 135
$\mathrm{HH}^{ 4 }$ = 90 0 0 0 0 90
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 85t + 260t^{2} + 85t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(68)
Coates et al. (2020) P1 x Q3
Coates–Kasprzyk–Prince (2015) CKP(111)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 2t - 1 \right) \left( 2t + 1 \right) \left( 100t^{3} - 12t^{2} - 6t - 1 \right) \left( 116t^{3} - 12t^{2} + 6t - 1 \right) \left( 1100898t^{7} - 1390544t^{6} + 2038500t^{5} - 419681t^{4} + 45360t^{3} - 7153t^{2} + 324t - 16 \right)D^{6} + 2 \left( 536357505600t^{15} - 806753596544t^{14} + 1135864253232t^{13} - 314858612288t^{12} - 116871249600t^{11} + 48942283800t^{10} - 8742697200t^{9} + 2038751776t^{8} - 330796629t^{7} + 26999440t^{6} + 4508730t^{5} - 1539012t^{4} + 152712t^{3} - 21971t^{2} + 810t - 32 \right)D^{5} + \left( 8939291760000t^{15} - 13571658607360t^{14} + 20564393837400t^{13} - 6110676866112t^{12} - 917636846544t^{11} + 466482589380t^{10} - 81982012032t^{9} + 15302866580t^{8} - 2110620618t^{7} + 262515344t^{6} - 71773236t^{5} + 8197527t^{4} - 626940t^{3} + 63209t^{2} - 1620t + 64 \right)D^{4} + 2 t ^{2} \left( 18772512696000t^{13} - 28707995872640t^{12} + 45868582993320t^{11} - 14068282877552t^{10} - 407555806176t^{9} + 496612421022t^{8} - 98780893848t^{7} + 21314441062t^{6} - 2033625393t^{5} + 179496360t^{4} - 19030464t^{3} + 1041142t^{2} - 88344t - 832 \right)D^{3} + 4 t ^{2} \left( 20739156883200t^{13} - 31890897923584t^{12} + 52934729077098t^{11} - 16462843447440t^{10} + 685071800388t^{9} + 194347971218t^{8} - 59410814580t^{7} + 18716401102t^{6} - 1479812166t^{5} + 95493028t^{4} - 7434963t^{3} + 234242t^{2} - 51408t - 128 \right)D^{2} + 8 t ^{3} \left( 11263507617600t^{12} - 17389864521344t^{11} + 29646864421902t^{10} - 9272021187716t^{9} + 773964552168t^{8} - 20340188716t^{7} - 15269860746t^{6} + 8637986976t^{5} - 658595664t^{4} + 34115223t^{3} - 1643436t^{2} - 73327t - 14580 \right)D + 48 t ^{3} \left( 766225008000t^{12} - 1186192280320t^{11} + 2058001150140t^{10} - 644949148684t^{9} + 69573328092t^{8} - 6736789952t^{7} - 331853094t^{6} + 521395356t^{5} - 40806405t^{4} + 1971765t^{3} - 63684t^{2} - 9911t - 540 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ1P1

|λ| = 6.762203 λ ≈ 6.762203; multiplicity 1; Jordan blocks 1 λ ≈ -4.381102 - 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ -4.381102 + 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3811016 - 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3811016 + 4.124189i; multiplicity 1; Jordan blocks 1 λ ≈ 2.762203; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ 2; multiplicity 1; Jordan blocks 1
Sector dimension
8
Spectral radius
$6.76220316$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.762203$11
$-4.381102 - 4.124189\mathrm{i}$11
$-4.381102 + 4.124189\mathrm{i}$11
$-0.3811016 - 4.124189\mathrm{i}$11
$-0.3811016 + 4.124189\mathrm{i}$11
$2.762203$11
$-2$11
$2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{3} + 6 \lambda^{2} + 12 \lambda - 100\right) \left(\lambda^{3} - 6 \lambda^{2} + 12 \lambda - 116\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda - 2\right) \left(\lambda^{3} + 6 \lambda^{2} + 12 \lambda - 100\right) \left(\lambda^{3} - 6 \lambda^{2} + 12 \lambda - 116\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 0 & 6 & 0 & 0 & 0 \\ 3 & 0 & 0 & 2 & 0 & 0 & 6 & 0 \\ 2 & 0 & 0 & 0 & 0 & 6 & 0 & 0 \\ 0 & 2 & 3 & 0 & 0 & 0 & 0 & 6 \\ 0 & 3 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$