Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-240-54-e

·

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^(w3) + E^(w4) + E^(2w1)
Bundle
O(0,1) + Q_1^v ot wedge^2 U_2^v ot U_3^v + O(2,0)
ZL62 label
44.324
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,4) \\ \det\mathcal{E} &\cong \mathcal{O}(2,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 240
$\chi(X, \mathrm{T}_X)$ = 1

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 240
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 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
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 1 0 x0 - 1 x1
$\mathrm{HH}^{ 3 }$ = x1 + 46 0 0 x1 - 5 51
$\mathrm{HH}^{ 4 }$ = 54 0 0 0 0 54
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 49t + 140t^{2} + 49t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(230)
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) \left( 5584t^{6} + 64t^{5} - 592t^{4} - 440t^{3} - 16t^{2} - 4t + 1 \right) \left( 472856040432t^{9} - 112631736960t^{8} - 263782734944t^{7} - 69992864976t^{6} + 1593950320t^{5} + 76447576t^{4} + 27289720t^{3} - 8035184t^{2} - 112561t - 780 \right)D^{6} + \left( 179549112824515584t^{16} - 9384133991696640t^{15} - 130959085854722048t^{14} - 57340332039095040t^{13} + 2243967919963008t^{12} + 6632821764747008t^{11} + 1682930473938944t^{10} + 89977212898864t^{9} - 6632980701120t^{8} + 1667772666448t^{7} + 689865750848t^{6} + 22094409824t^{5} + 1106078632t^{4} - 114507640t^{3} + 32290496t^{2} + 337683t + 1560 \right)D^{5} + \left( 1214596939695252480t^{16} - 131179647883348800t^{15} - 909558580967559680t^{14} - 378231035345038464t^{13} - 5084060172853696t^{12} + 19342058707882176t^{11} + 3475995015952480t^{10} - 60726582557296t^{9} - 31207790932736t^{8} - 5234795480036t^{7} - 188033290480t^{6} + 135710770256t^{5} + 1111372912t^{4} + 133323552t^{3} - 29181084t^{2} - 269283t - 780 \right)D^{4} + t \left( 4171876445040215040t^{15} - 615819889588694400t^{14} - 3223502195773561856t^{13} - 1329353679374256128t^{12} - 57225020534597120t^{11} + 26937233785943680t^{10} + 3336126940055360t^{9} - 338444696977520t^{8} + 5336707097280t^{7} + 10705205885864t^{6} + 3225627713152t^{5} + 239718628224t^{4} + 5513111384t^{3} + 83889160t^{2} + 740144t + 481 \right)D^{3} + 4 t ^{2} \left( 1911669965955136512t^{14} - 333491744078956080t^{13} - 1522250761983844736t^{12} - 632317781275722080t^{11} - 34617893278463344t^{10} + 4252499378026960t^{9} + 468033095939832t^{8} - 14241702572080t^{7} + 24334967710480t^{6} + 5845327222833t^{5} + 1205214710496t^{4} + 62158307296t^{3} + 1061210880t^{2} + 21933906t + 169440 \right)D^{2} + 8 t ^{2} \left( 881902995343944192t^{14} - 169058355996629520t^{13} - 720925364827125376t^{12} - 303439089384935168t^{11} - 17550994885778576t^{10} + 211562263510576t^{9} + 155130038527368t^{8} + 42651702773168t^{7} + 15181331497664t^{6} + 2795441870651t^{5} + 449745677144t^{4} + 17265597658t^{3} + 261491848t^{2} + 5465906t + 37440 \right)D + 32 t ^{2} \left( 79212843893168640t^{14} - 16024062004801200t^{13} - 66114484078551168t^{12} - 28210876886112144t^{11} - 1627955261853328t^{10} - 58535437614304t^{9} + 19825688730952t^{8} + 6463704935328t^{7} + 1488050971384t^{6} + 252758429241t^{5} + 33379224352t^{4} + 1012874113t^{3} + 15884154t^{2} + 277381t + 1560 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 5)

|λ| = 10.24693 λ ≈ 10.24693; multiplicity 1; Jordan blocks 1 λ ≈ -2.319438 - 5.924335i; multiplicity 1; Jordan blocks 1 λ ≈ -2.319438 + 5.924335i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 1; Jordan blocks 1 λ ≈ -1.773015 - 1.950202i; multiplicity 1; Jordan blocks 1 λ ≈ -1.773015 + 1.950202i; multiplicity 1; Jordan blocks 1 λ ≈ 1.937976; multiplicity 1; Jordan blocks 1 λ ≈ 0; multiplicity 1; Jordan blocks 1
Sector dimension
8
Spectral radius
$10.2469291$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$10.24693$11
$-2.319438 - 5.924335\mathrm{i}$11
$-2.319438 + 5.924335\mathrm{i}$11
$-4$11
$-1.773015 - 1.950202\mathrm{i}$11
$-1.773015 + 1.950202\mathrm{i}$11
$1.937976$11
$0$11

Characteristic polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right) \left(\lambda^{6} - 4 \lambda^{5} - 16 \lambda^{4} - 440 \lambda^{3} - 592 \lambda^{2} + 64 \lambda + 5584\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\lambda \left(\lambda + 4\right) \left(\lambda^{6} - 4 \lambda^{5} - 16 \lambda^{4} - 440 \lambda^{3} - 592 \lambda^{2} + 64 \lambda + 5584\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 2 & 12 & 6 & 16 & 0 & 0 \\ 1 & 0 & 0 & 2 & 0 & 6 & 0 & 0 \\ 2 & 2 & 0 & 4 & 4 & 6 & 6 & 16 \\ 0 & 2 & 5 & 4 & 4 & 4 & 0 & 6 \\ 0 & 1 & -4 & -4 & -4 & -2 & 2 & 0 \\ 0 & 0 & 0 & 6 & 5 & 0 & 0 & 2 \\ 0 & 0 & 0 & -1 & -3 & 2 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0\end{pmatrix}\notag\end{equation}$$