Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-88-25-b

·

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

Defining data
Ambient
Fl(1, 5, 8)
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_7ω1ω2ω3ω4ω5ω6ω7
Bundle
4 copies of E^(w7) + 2 copies of E^(2w1) + E^(2w5)
Bundle
4 copies of wedge^2 Q_1^v ot wedge^4 U_2^v ot U_3^v + 2 copies of O(2,0) + O(0,2)
ZL62 label
70G.6666B77
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5,7) \\ \det\mathcal{E} &\cong \mathcal{O}(4,6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 25
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 88
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 88
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 128
$\mathrm{c}_1\mathrm{c}_3$ = 16
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 3 0
0 1 1 0
0 2 32 2 0
0 1 1 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x2 + 28 0 x0 + 28 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 17 0 0 x1 - x2 - 10 2 25
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 20t + 46t^{2} + 20t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(666)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(669)

Quantum differential equation

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

$$\begin{equation} L_X=\left( 12t - 1 \right) \left( 28t - 1 \right) \left( 8t + 1 \right) ^{2} \left( 4t + 1 \right) ^{4} \left( 30116864t^{5} + 21386496t^{4} + 3403264t^{3} + 63616t^{2} - 4812t + 11 \right)D^{6} + 2 \left( 8t + 1 \right) \left( 4t + 1 \right) ^{3} \left( 3400073478144t^{9} + 2808908152832t^{8} + 616656945152t^{7} + 27371601920t^{6} - 2661000192t^{5} - 62393856t^{4} - 224064t^{3} - 219008t^{2} + 7240t - 11 \right)D^{5} + \left( 4t + 1 \right) ^{2} \left( 1813372521676800t^{11} + 1975091640401920t^{10} + 743171685941248t^{9} + 122173267427328t^{8} + 7925577666560t^{7} - 12462144512t^{6} - 16081268480t^{5} + 51941952t^{4} + 38447024t^{3} + 369404t^{2} - 8562t + 11 \right)D^{4} + 2 t \left( 4t + 1 \right) \left( 15232329182085120t^{11} + 19059003212431360t^{10} + 8814670109736960t^{9} + 1953762502049792t^{8} + 212844690784256t^{7} + 9111435558912t^{6} - 117952054272t^{5} - 15982267392t^{4} - 17908160t^{3} + 6377536t^{2} - 106416t - 25 \right)D^{3} + 4 t ^{2} \left( 67312388004642816t^{11} + 96520774566805504t^{10} + 53813144148770816t^{9} + 15249565632888832t^{8} + 2342957881524224t^{7} + 184609350123520t^{6} + 5091884527616t^{5} - 169457286656t^{4} - 10516317824t^{3} - 18368256t^{2} + 452660t - 31801 \right)D^{2} + 32 t ^{2} \left( 9139397509251072t^{11} + 12654462436900864t^{10} + 6708394046849024t^{9} + 1783765743960064t^{8} + 252946872086528t^{7} + 17821799727104t^{6} + 370726062336t^{5} - 20179610368t^{4} - 940493936t^{3} - 1929952t^{2} - 78626t - 1265 \right)D + 192 t ^{2} \left( 621727721717760t^{11} + 840132743659520t^{10} + 429375458246656t^{9} + 108831798444032t^{8} + 14486824202240t^{7} + 927421314048t^{6} + 13795836928t^{5} - 1225835200t^{4} - 46301392t^{3} - 119316t^{2} - 6180t - 33 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 5, 8)

|λ| = 28 λ ≈ 28; multiplicity 1; Jordan blocks 1 λ ≈ 12; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 2; Jordan blocks 1, 1 λ ≈ -4; multiplicity 5; Jordan blocks 1, 1, 3 2 5
Sector dimension
9
Spectral radius
$28$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$28$11
$12$11
$-8$21, 1
$-4$51, 1, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{5} \left(\lambda + 8\right)^{2} \left(\lambda - 12\right) \left(\lambda - 28\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda + 8\right) \left(\lambda - 12\right) \left(\lambda - 28\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 56 & 16 & 240 & 480 & 0 & 320 & 1984 & 3840 \\ 1 & 4 & 2 & 32 & 72 & -4 & 0 & 240 & 320 \\ 1 & 4 & -2 & 56 & 72 & 20 & 240 & 528 & 1664 \\ 0 & 1 & 1 & 2 & 8 & 2 & 36 & 72 & 240 \\ 0 & 1 & 0 & 2 & 0 & 0 & -4 & 16 & 0 \\ 0 & 0 & 1 & 4 & 4 & -2 & 40 & 56 & 240 \\ 0 & 0 & 0 & 1 & 0 & 1 & 2 & 8 & 40 \\ 0 & 0 & 0 & 1 & 2 & 0 & 2 & 0 & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$