Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-96-26-a

·

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

Defining data
Ambient
Fl(1, 3, 5)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w4) + E^(2w1) + E^(2w3)
Bundle
Q_1^v ot wedge^2 U_2^v ot U_3^v + O(2,0) + O(0,2)
ZL62 label
44.364
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 26
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 96
$\chi(X, \mathrm{T}_X)$ = -13

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 96
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 32
$\mathrm{c}_4$ = 8
Hodge diamond
1
00
0 2 0
0 2 2 0
0 0 10 0 0
0 2 2 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 + 13 0 x0 + 13 x2
$\mathrm{HH}^{ 3 }$ = x1 + 5 0 0 x1 5
$\mathrm{HH}^{ 4 }$ = x1 - x2 - 1 0 0 x1 - x2 - 27 0 26
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 21t + 52t^{2} + 21t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(616)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

Quantum differential equation

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

$$\begin{equation} L_X=\left( 4t + 1 \right) ^{2} \left( 1600t^{3} + 272t^{2} + 8t - 1 \right) \left( 745600000t^{7} + 752064000t^{6} + 248889600t^{5} + 33807040t^{4} + 11306880t^{3} + 1577968t^{2} + 11455t + 60 \right)D^{6} + \left( 4t + 1 \right) \left( 52490240000000t^{11} + 67015884800000t^{10} + 31935563776000t^{9} + 7526777139200t^{8} + 1867993318400t^{7} + 447471303680t^{6} + 54736258048t^{5} + 2431377152t^{4} + 52038672t^{3} + 6291532t^{2} + 33885t + 120 \right)D^{5} + \left( 940052480000000t^{12} + 1355561984000000t^{11} + 748064485376000t^{10} + 210141605478400t^{9} + 54035229020160t^{8} + 14347839226880t^{7} + 2295814827520t^{6} + 166403496128t^{5} + 4080097600t^{4} - 6345560t^{3} - 5782324t^{2} - 30285t - 60 \right)D^{4} + t \left( 2195046400000000t^{11} + 3068167168000000t^{10} + 1575406305280000t^{9} + 401825972838400t^{8} + 117293250232320t^{7} + 32288456115200t^{6} + 4529044440320t^{5} + 259448519296t^{4} + 4482727360t^{3} + 55949040t^{2} + 421600t + 175 \right)D^{3} + 8 t ^{2} \left( 352519680000000t^{10} + 485437676800000t^{9} + 235887049344000t^{8} + 55826093811200t^{7} + 18898631045120t^{6} + 5190992084160t^{5} + 644236514688t^{4} + 29331976970t^{3} + 417358400t^{2} + 7639595t + 43200 \right)D^{2} + 16 t ^{2} \left( 118103040000000t^{10} + 161762684800000t^{9} + 75284263616000t^{8} + 16789872556800t^{7} + 6580592929280t^{6} + 1779207950880t^{5} + 200052559568t^{4} + 7374762923t^{3} + 102338160t^{2} + 1838385t + 8400 \right)D + 160 t ^{2} \left( 3220992000000t^{10} + 4411369600000t^{9} + 1983562406400t^{8} + 421445710080t^{7} + 188718553600t^{6} + 50125554912t^{5} + 5232492432t^{4} + 160464781t^{3} + 2496002t^{2} + 34987t + 120 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 22.91613 λ ≈ 22.91613; multiplicity 1; Jordan blocks 1 λ ≈ -7.458063 - 3.767907i; multiplicity 1; Jordan blocks 1 λ ≈ -7.458063 + 3.767907i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 2; Jordan blocks 2 λ ≈ 0; multiplicity 3; Jordan blocks 1, 2 2 3
Sector dimension
8
Spectral radius
$22.9161262$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22.91613$11
$-7.458063 - 3.767907\mathrm{i}$11
$-7.458063 + 3.767907\mathrm{i}$11
$-4$22
$0$31, 2

Characteristic polynomial

$$\begin{equation}\lambda^{3} \left(\lambda + 4\right)^{2} \left(\lambda^{3} - 8 \lambda^{2} - 272 \lambda - 1600\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda^{2} \left(\lambda + 4\right)^{2} \left(\lambda^{3} - 8 \lambda^{2} - 272 \lambda - 1600\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 20 & 20 & 168 & 96 & 544 & 256 & 1600 \\ 1 & 0 & 4 & 20 & 16 & 96 & 48 & 256 \\ 1 & 2 & 0 & 36 & 20 & 96 & 48 & 288 \\ 0 & 1 & 3 & 4 & 4 & 36 & 16 & 96 \\ 0 & 1 & -2 & 0 & -4 & -16 & -12 & -24 \\ 0 & 0 & 0 & 4 & 3 & 4 & 4 & 20 \\ 0 & 0 & 0 & -1 & -2 & -2 & -4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0\end{pmatrix}\notag\end{equation}$$