Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-240-54-f

·

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

Defining data
Ambient
Fl(1, 2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w5) + E^(3w2)
Bundle
wedge^3 Q_1^v ot U_2^v ot U_3^v + O(0,3)
ZL62 label
52.4B
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,5) \\ \det\mathcal{E} &\cong \mathcal{O}(0,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 54
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 240
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 240
$\mathrm{c}_1^2\mathrm{c}_2$ = 156
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = -12
Hodge diamond
1
00
0 2 0
0 5 5 0
0 0 2 0 0
0 5 5 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 + 4 0 x0 + 4 x2
$\mathrm{HH}^{ 3 }$ = x1 + 65 0 0 x1 65
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 49 0 0 x1 - x2 - 5 0 54
$\mathrm{HH}^{ 5 }$ = 1 0 0 1 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(127)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(169)

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) ^{2} \left( 2t + 1 \right) ^{2} \left( 19667t^{4} + 972t^{3} + 8t^{2} - 27t - 1 \right) \left( 292192764771600t^{9} + 45193448223936t^{8} + 3741242699340t^{7} + 528348567568t^{6} - 3638898576t^{5} - 14747904992t^{4} + 1367347014t^{3} + 19473592t^{2} + 4323t + 3240 \right)D^{6} + \left( 2t - 1 \right) \left( 2t + 1 \right) \left( 482710628800096804800t^{15} + 97528893047716813056t^{14} - 18679135197717743616t^{13} - 4606781263824372288t^{12} - 614825798265182844t^{11} - 184259777654609584t^{10} - 11064308968369368t^{9} + 2106028539741580t^{8} - 275413525588158t^{7} - 19200002104560t^{6} + 543027708723t^{5} + 49075604422t^{4} - 6874924146t^{3} - 78114607t^{2} + 161991t - 6480 \right)D^{5} + \left( 16090354293336560160000t^{17} + 3224592336447965076480t^{16} - 2724222353819229561600t^{15} - 565284260339658346240t^{14} - 25761323493404916768t^{13} - 3803674812680683648t^{12} + 798175795632921108t^{11} + 420870572003046976t^{10} - 37240628701435416t^{9} - 13961872146505720t^{8} + 883192722340866t^{7} + 37199325606496t^{6} - 190881041067t^{5} + 306161277812t^{4} - 5136110658t^{3} - 75932890t^{2} - 158289t - 3240 \right)D^{4} + t \left( 67579488032013552672000t^{16} + 13456264288110888637440t^{15} - 6428682081941303240640t^{14} - 1346976510757178895360t^{13} - 184985758614070899504t^{12} - 38047391315670944384t^{11} + 109899676795012308t^{10} + 1228253103645155472t^{9} - 80855725667713968t^{8} - 24737385814642224t^{7} + 2041862030069502t^{6} + 78259315873640t^{5} + 842688440451t^{4} + 14972826276t^{3} + 340972794t^{2} + 1389312t + 243 \right)D^{3} + 2 t ^{2} \left( 74659243921081639142400t^{15} + 14792412567425956821504t^{14} - 3687222890797467559776t^{13} - 796460340696101475584t^{12} - 153473748361517163672t^{11} - 36424194640105207456t^{10} - 111484441361937996t^{9} + 980137171623147520t^{8} - 80744990981385108t^{7} - 13390349969955644t^{6} + 1050282097940196t^{5} + 31145341464384t^{4} + 298834064775t^{3} + 10464965704t^{2} + 204710460t + 584640 \right)D^{2} + 4 t ^{2} \left( 40547692819208131603200t^{15} + 8004535314367220365824t^{14} - 870279060813317332704t^{13} - 204777183051286002048t^{12} - 53648587371555352584t^{11} - 15434167794113418528t^{10} + 118819274382305964t^{9} + 452857944160738264t^{8} - 40596501392511084t^{7} - 3874618508148728t^{6} + 245076077660532t^{5} + 5317723929928t^{4} + 50532682797t^{3} + 3698551908t^{2} + 63542628t + 133920 \right)D + 24 t ^{2} \left( 2758346450286267456000t^{15} + 543180764660562846720t^{14} - 13520934668192806080t^{13} - 4773236242448165760t^{12} - 2309275261454482656t^{11} - 849004968097573824t^{10} + 18798097297742748t^{9} + 28561899533123176t^{8} - 2643039502312896t^{7} - 161162834897108t^{6} + 6362741020770t^{5} + 54054427168t^{4} + 862364901t^{3} + 176319468t^{2} + 2707122t + 4320 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 6)

|λ| = 26.96301 λ ≈ -26.96301; multiplicity 1; Jordan blocks 1 λ ≈ 9.321945; multiplicity 1; Jordan blocks 1 λ ≈ -4.679466 - 7.506582i; multiplicity 1; Jordan blocks 1 λ ≈ -4.679466 + 7.506582i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 λ ≈ 2; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
8
Spectral radius
$26.9630136$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$-26.96301$11
$9.321945$11
$-4.679466 - 7.506582\mathrm{i}$11
$-4.679466 + 7.506582\mathrm{i}$11
$-2$22
$2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right)^{2} \left(\lambda^{4} + 27 \lambda^{3} - 8 \lambda^{2} - 972 \lambda - 19667\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda - 2\right)^{2} \left(\lambda^{4} + 27 \lambda^{3} - 8 \lambda^{2} - 972 \lambda - 19667\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 0 & 18 & 0 & 0 & 72 & 0 \\ 2 & 0 & 6 & 0 & -108 & 0 & 756 & 72 \\ 1 & 0 & -6 & 2 & 108 & 45 & -756 & -72 \\ 0 & 3 & 2 & 0 & 15 & 0 & -108 & 0 \\ 0 & 0 & 1 & 0 & -15 & 2 & 108 & 18 \\ 0 & 0 & 0 & 3 & 2 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & -6 & 2 \\ 0 & 0 & 0 & 0 & 0 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$