Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-27-12-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ5P5
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_6ω1ω2ω3ω4ω5ω6
Bundle
E^(3w2) + E^(w1 + 2w2)
Bundle
O_2(3) + O_1(1) box O_2(2)
ZL62 label
150.Dx301
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;6) \\ \det\mathcal{E} &\cong \mathcal{O}(1;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 12
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 27
$\chi(X, \mathrm{T}_X)$ = -58

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 27
$\mathrm{c}_1^2\mathrm{c}_2$ = 78
$\mathrm{c}_2^2$ = 192
$\mathrm{c}_1\mathrm{c}_3$ = -42
$\mathrm{c}_4$ = 99
Hodge diamond
1
00
0 2 0
0 0 0 0
0 7 79 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 + 58 0 x0 + 58 0
$\mathrm{HH}^{ 3 }$ = x1 0 0 x1 0
$\mathrm{HH}^{ 4 }$ = x1 + 46 0 0 x1 + 34 0 12
$\mathrm{HH}^{ 5 }$ = 6 0 0 6 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 7t + 11t^{2} + 7t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(738)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(729)

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) ^{2} \left( 13t + 1 \right) ^{3} \left( 109863t^{3} + 5940t^{2} - 180t + 1 \right) \left( 9561864211069014t^{9} + 4605302594364264t^{8} + 904792205216718t^{7} + 91391060193183t^{6} + 4734989638632t^{5} + 85013410977t^{4} - 2686079766t^{3} - 139163563t^{2} - 1529622t + 675 \right)D^{6} + 2 \left( 12t + 1 \right) \left( 13t + 1 \right) ^{2} \left( 1720710953850265789364316t^{14} + 1089021267324480113001729t^{13} + 296941832806072392527589t^{12} + 45232079500852101335907t^{11} + 4136815120565726635374t^{10} + 221206838567141140533t^{9} + 5419807074117328464t^{8} - 67590842495271111t^{7} - 8160435554574105t^{6} - 194322909674682t^{5} - 1329246903198t^{4} + 10497282740t^{3} + 275251259t^{2} + 2241108t - 675 \right)D^{5} + \left( 13t + 1 \right) \left( 4473848480010691052347221600t^{16} + 3322700921795897136192881040t^{15} + 1092519015645946734941828382t^{14} + 208418791109916008494255500t^{13} + 25303831387871353478733312t^{12} + 2000974482907759799873901t^{11} + 98825782412685939270660t^{10} + 2421709591949402267034t^{9} - 24732323623796016267t^{8} - 3938643719994110508t^{7} - 127076105677148244t^{6} - 1854036129822558t^{5} - 9387443809032t^{4} + 14379703727t^{3} - 377808446t^{2} - 2875464t + 675 \right)D^{4} + 2 t \left( 122136063504291865729079149680t^{16} + 95263181516849430204782115480t^{15} + 33164465735209106592508667895t^{14} + 6771361551861516818986674585t^{13} + 893298986470445882838296994t^{12} + 78591797314528785584753781t^{11} + 4519183813115957253467550t^{10} + 149291556602768261489367t^{9} + 844055928083179905489t^{8} - 166141854688092777360t^{7} - 8191988584459582221t^{6} - 188096625699128307t^{5} - 2210412769504184t^{4} - 10535916661605t^{3} - 1700005224t^{2} - 132492821t - 9675 \right)D^{3} + 4 t ^{2} \left( 134931270157122442138792203456t^{15} + 101150799135397085863758076848t^{14} + 33691782647254967093002437306t^{13} + 6544540335083220015546908934t^{12} + 814923212732225714123308809t^{11} + 66798696090923837780444172t^{10} + 3480956591400897371966385t^{9} + 94315279023336199501629t^{8} - 646814794787003434896t^{7} - 161837642875925445435t^{6} - 6387093412407357282t^{5} - 127356282426300645t^{4} - 1300398389023661t^{3} - 5122566466493t^{2} + 1512892299t - 32732226 \right)D^{2} + 24 t ^{2} \left( 24427212700858373145815829936t^{15} + 17768869326346890216619721736t^{14} + 5720241211534076626545984513t^{13} + 1068522902352580416171080391t^{12} + 127030494290252884080269247t^{11} + 9819873698010979562118900t^{10} + 469127391334851147283827t^{9} + 10254049730094450017097t^{8} - 231744945029829804090t^{7} - 25222420799418342609t^{6} - 858735128630084097t^{5} - 15281535571243636t^{4} - 137688357197221t^{3} - 454589456967t^{2} + 18100893t - 1355040 \right)D + 144 t ^{2} \left( 1661715149718256676586110880t^{15} + 1183814881733549464214428080t^{14} + 372039199057793917699423686t^{13} + 67564054051192679430521682t^{12} + 7761993817848346416018978t^{11} + 573683043655305337067496t^{10} + 25522716238159156856541t^{9} + 446538948694980694695t^{8} - 18886854624476530932t^{7} - 1511125072728012009t^{6} - 46453571582876928t^{5} - 756546517719969t^{4} - 6145902558998t^{3} - 17551078547t^{2} - 11879355t - 33300 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ5P5

|λ| = 125.8779 λ ≈ 125.8779; multiplicity 1; Jordan blocks 1 λ ≈ 67.12441; multiplicity 1; Jordan blocks 1 λ ≈ -13.00234; multiplicity 1; Jordan blocks 1 λ ≈ -13; multiplicity 3; Jordan blocks 3 λ ≈ -12; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
8
Spectral radius
$125.87793$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$125.8779$11
$67.12441$11
$-13.00234$11
$-13$33
$-12$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 12\right)^{2} \left(\lambda + 13\right)^{3} \left(\lambda^{3} - 180 \lambda^{2} + 5940 \lambda + 109863\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 12\right)^{2} \left(\lambda + 13\right)^{3} \left(\lambda^{3} - 180 \lambda^{2} + 5940 \lambda + 109863\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 48 & 840 & 4788 & 49698 & 357408 & 1121472 & 8334720 \\ 1 & -1 & 18 & 816 & 1704 & 47736 & 61668 & 942768 \\ 1 & 2 & 30 & 168 & 2484 & 14130 & 54801 & 357408 \\ 0 & 1 & 1 & 29 & 51 & 2400 & 2496 & 47304 \\ 0 & 0 & 1 & 2 & 30 & 168 & 900 & 4788 \\ 0 & 0 & 0 & 1 & 1 & 29 & 33 & 816 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 48 \\ 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$