Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-36-14-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ6P6
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_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
2 copies of E^(2w2) + E^(w1 + 2w2)
Bundle
2 copies of O_2(2) + O_1(1) box O_2(2)
ZL62 label
1600.88x301
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;7) \\ \det\mathcal{E} &\cong \mathcal{O}(1;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 14
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 36
$\chi(X, \mathrm{T}_X)$ = -52

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 36
$\mathrm{c}_1^2\mathrm{c}_2$ = 84
$\mathrm{c}_2^2$ = 180
$\mathrm{c}_1\mathrm{c}_3$ = -28
$\mathrm{c}_4$ = 92
Hodge diamond
1
00
0 2 0
0 0 0 0
0 7 72 7 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 + 52 0 x0 + 52 0
$\mathrm{HH}^{ 3 }$ = x1 0 0 x1 0
$\mathrm{HH}^{ 4 }$ = x1 + 41 0 0 x1 + 27 0 14
$\mathrm{HH}^{ 5 }$ = 2 0 0 2 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 9t + 16t^{2} + 9t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(733)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(725)

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( 8t + 1 \right) ^{2} \left( 9t + 1 \right) ^{3} \left( 24144t^{3} + 1664t^{2} - 104t + 1 \right) \left( 310796253037332t^{9} + 231208235111664t^{8} + 70251990160152t^{7} + 11181644631259t^{6} + 975085824412t^{5} + 41840066627t^{4} + 343799776t^{3} - 31446051t^{2} - 668612t + 441 \right)D^{6} + 2 \left( 8t + 1 \right) \left( 9t + 1 \right) ^{2} \left( 5672921738400007918848t^{14} + 5446795145713647071904t^{13} + 2248618575107457496224t^{12} + 520813260911864536128t^{11} + 73607381396330725914t^{10} + 6389845680846689602t^{9} + 313865583786421780t^{8} + 5641345299300602t^{7} - 186910391179462t^{6} - 9898728532769t^{5} - 94731340761t^{4} + 1003255300t^{3} + 54972148t^{2} + 987483t - 441 \right)D^{5} + \left( 9t + 1 \right) \left( 6807506086080009502617600t^{16} + 7659801462553500974238720t^{15} + 3810578020795436243140944t^{14} + 1103040431360640800461728t^{13} + 205182088596267641778672t^{12} + 25455492719650553129484t^{11} + 2099039353370460306040t^{10} + 108640203647334444712t^{9} + 2818168454595062073t^{8} - 17594861701453014t^{7} - 3412811535881588t^{6} - 78016467651030t^{5} - 96588997590t^{4} + 11890024950t^{3} - 68052432t^{2} - 1249710t + 441 \right)D^{4} + 4 t \left( 64330932513456089799736320t^{16} + 75941677215506726848085760t^{15} + 39962680679149501820774040t^{14} + 12367302496756610780158920t^{13} + 2494681449440698637922672t^{12} + 342430776783343256355243t^{11} + 32233707309165466937469t^{10} + 2019449612394974427736t^{9} + 75209760741176377620t^{8} + 870849651310109545t^{7} - 57095595855120357t^{6} - 2960894977186660t^{5} - 55611018322200t^{4} - 345189522289t^{3} + 539273749t^{2} - 17485755t - 1813 \right)D^{3} + 8 t ^{2} \left( 71070363538675299207327744t^{15} + 80861944736696019888462336t^{14} + 40823187638300711741898630t^{13} + 12061973971925316938165634t^{12} + 2309857551626457497023458t^{11} + 298735242928768812357306t^{10} + 26185632877576859507814t^{9} + 1492832457164141267399t^{8} + 47140350498134691297t^{7} + 116078801670069846t^{6} - 54515039085523930t^{5} - 2147824970791523t^{4} - 34669387092713t^{3} - 190852828386t^{2} + 129335220t - 4395132 \right)D^{2} + 16 t ^{2} \left( 38598559508073653879841792t^{15} + 42708268150661414364132096t^{14} + 20883600985974601565722050t^{13} + 5950940482856387070371718t^{12} + 1093408218080337600796086t^{11} + 134724082671619717615083t^{10} + 11123128641213101162373t^{9} + 583149086483750757949t^{8} + 15521996860402748251t^{7} - 122477711194703648t^{6} - 24720758116807812t^{5} - 811909822603745t^{4} - 11445289685579t^{3} - 53965187353t^{2} - 28818601t - 555660 \right)D + 32 t ^{2} \left( 7877257042464010995886080t^{15} + 8549399437240369010480640t^{14} + 4087465629991857951108012t^{13} + 1134844151561853291665172t^{12} + 202279772682300285969372t^{11} + 24031714683812674751262t^{10} + 1893688530553487588406t^{9} + 92624340745292621410t^{8} + 2083369389340968380t^{7} - 40522690509947999t^{6} - 4321378864330409t^{5} - 125180868287368t^{4} - 1585782068742t^{3} - 6612127305t^{2} - 13681339t - 40572 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ6P6

|λ| = 79.10707 λ ≈ 79.10707; multiplicity 1; Jordan blocks 1 λ ≈ 33.8969; multiplicity 1; Jordan blocks 1 λ ≈ -9.003969; multiplicity 1; Jordan blocks 1 λ ≈ -9; multiplicity 3; Jordan blocks 3 λ ≈ -8; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$79.1070731$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$79.10707$11
$33.8969$11
$-9.003969$11
$-9$33
$-8$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 8\right)^{2} \left(\lambda + 9\right)^{3} \left(\lambda^{3} - 104 \lambda^{2} + 1664 \lambda + 24144\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 8\right)^{2} \left(\lambda + 9\right)^{3} \left(\lambda^{3} - 104 \lambda^{2} + 1664 \lambda + 24144\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 32 & 336 & 1848 & 12300 & 83072 & 201472 & 1315200 \\ 1 & -1 & 12 & 320 & 656 & 11568 & 15672 & 159936 \\ 1 & 2 & 16 & 96 & 880 & 4848 & 13992 & 83072 \\ 0 & 1 & 1 & 15 & 28 & 832 & 912 & 11376 \\ 0 & 0 & 1 & 2 & 16 & 96 & 368 & 1848 \\ 0 & 0 & 0 & 1 & 1 & 15 & 20 & 320 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 32 \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$