Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-45-16-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ5P5
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6node 7: fundamental weight ω_7ω1ω2ω3ω4ω5ω6ω7
Bundle
E^(3w3) + 2 copies of E^(w1 + w3)
Bundle
O_2(3) + 2 copies of O_1(1) box O_2(1)
ZL62 label
2050.Gw1w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;6) \\ \det\mathcal{E} &\cong \mathcal{O}(2;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 16
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 45
$\chi(X, \mathrm{T}_X)$ = -44

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 45
$\mathrm{c}_1^2\mathrm{c}_2$ = 90
$\mathrm{c}_2^2$ = 162
$\mathrm{c}_1\mathrm{c}_3$ = -18
$\mathrm{c}_4$ = 63
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 51 3 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 + x2 + 44 0 x0 + 44 x2
$\mathrm{HH}^{ 3 }$ = x1 + 2 0 0 x1 2
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 22 0 0 x1 - x2 + 6 0 16
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 11t + 21t^{2} + 11t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(718)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(712)

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( 57t - 1 \right) \left( 6t + 1 \right) ^{2} \left( 7t + 1 \right) ^{4} \left( 252t^{2} - 6t + 1 \right) \left( 1170391420800t^{8} + 949732366464t^{7} + 309552864624t^{6} + 50498557056t^{5} + 4286496006t^{4} + 197489832t^{3} + 7375215t^{2} + 251275t - 1000 \right)D^{6} + \left( 6t + 1 \right) \left( 7t + 1 \right) ^{3} \left( 14827745088903398400t^{13} + 14546453193375151104t^{12} + 6016593408303397248t^{11} + 1352102956769924352t^{10} + 180120971474372880t^{9} + 15060847650759072t^{8} + 875337713333496t^{7} + 40402755840690t^{6} + 1433255486652t^{5} + 43715967957t^{4} + 1482053742t^{3} + 11323075t^{2} + 1035100t - 3000 \right)D^{5} + \left( 7t + 1 \right) ^{2} \left( 5189710781116189440000t^{15} + 6288205619454613263360t^{14} + 3331317781711509440256t^{13} + 1007800331507019646848t^{12} + 191791859372987410656t^{11} + 24050287247027232240t^{10} + 2058831626612536224t^{9} + 127111076478852516t^{8} + 6123235236179814t^{7} + 230301335569680t^{6} + 5374067186145t^{5} + 27100413918t^{4} - 1669209806t^{3} - 50212330t^{2} - 769825t + 2000 \right)D^{4} + t \left( 7t + 1 \right) \left( 152577496964815969536000t^{15} + 197213074148636340802560t^{14} + 112641134651620484232960t^{13} + 37251069109642812848256t^{12} + 7885289891918271440160t^{11} + 1121791039834618602288t^{10} + 110798461542842422416t^{9} + 7905607006166912124t^{8} + 433317331968793830t^{7} + 19053127608714918t^{6} + 618598464131313t^{5} + 11265730719816t^{4} + 34844906156t^{3} - 2127445226t^{2} - 32794075t + 38200 \right)D^{3} + 2 t \left( 1179932643194576831078400t^{16} + 1637722587884672600220672t^{15} + 1014240960867234086880768t^{14} + 368239691879505137228544t^{13} + 86895938515965741130752t^{12} + 14023051237638810214848t^{11} + 1597017566207800876608t^{10} + 132319805068843331472t^{9} + 8328982750034837868t^{8} + 416907319931344386t^{7} + 16289413089755199t^{6} + 421987052338185t^{5} + 4527039981420t^{4} - 45552436682t^{3} - 1368805859t^{2} - 5612545t + 5500 \right)D^{2} + 8 t ^{2} \left( 320412743626113536025600t^{15} + 433591678616532984124416t^{14} + 260901230911654026863232t^{13} + 91632229784806781827008t^{12} + 20814551668645875119952t^{11} + 3218474320339495133448t^{10} + 350295443365651222320t^{9} + 27813902184332000334t^{8} + 1694596088961292419t^{7} + 82549012563970659t^{6} + 3052800263528874t^{5} + 68179689453222t^{4} + 359996405475t^{3} - 14436734540t^{2} - 219714055t - 162400 \right)D + 2592 t ^{3} \left( 403644171864592512000t^{14} + 536746157352631941120t^{13} + 316500694256398313088t^{12} + 108545851222482377088t^{11} + 23979932453850344496t^{10} + 3592359727566095304t^{9} + 378017667448958880t^{8} + 29094093956370234t^{7} + 1732136673185895t^{6} + 82583817233865t^{5} + 2907347054794t^{4} + 56681250570t^{3} + 46640175t^{2} - 15614500t - 178125 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ5P5

|λ| = 57 λ ≈ 57; multiplicity 1; Jordan blocks 1 λ ≈ 3 - 15.58846i; multiplicity 1; Jordan blocks 1 λ ≈ 3 + 15.58846i; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 4; Jordan blocks 2, 2 λ ≈ -6; multiplicity 2; Jordan blocks 2 4 2
Sector dimension
9
Spectral radius
$57$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$57$11
$3 - 15.58846\mathrm{i}$11
$3 + 15.58846\mathrm{i}$11
$-7$42, 2
$-6$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda + 7\right)^{4} \left(\lambda - 57\right) \left(\lambda^{2} - 6 \lambda + 252\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda + 7\right)^{2} \left(\lambda - 57\right) \left(\lambda^{2} - 6 \lambda + 252\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 36 & 144 & 1548 & 450 & 2664 & 38376 & 14040 & 290520 \\ 1 & -1 & 12 & 132 & 12 & 300 & 2484 & 756 & 20592 \\ 1 & 2 & 9 & 114 & 42 & 210 & 3492 & 1368 & 28080 \\ 0 & 1 & 1 & 8 & 0 & 30 & 168 & 30 & 1296 \\ 0 & 1 & 0 & 12 & -1 & 6 & 288 & 108 & 1836 \\ 0 & 0 & 1 & 2 & 1 & 0 & 96 & 42 & 900 \\ 0 & 0 & 0 & 1 & 0 & 3 & 3 & 2 & 72 \\ 0 & 0 & 0 & 1 & 1 & -3 & 24 & 5 & 144 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$