Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-41-15-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ4P4
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_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w1 + 2w3)
Bundle
2 copies of O_1(1) box O_2(2)
ZL62 label
2040.x311x311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 15
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 41
$\chi(X, \mathrm{T}_X)$ = -54

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 41
$\mathrm{c}_1^2\mathrm{c}_2$ = 86
$\mathrm{c}_2^2$ = 184
$\mathrm{c}_1\mathrm{c}_3$ = -26
$\mathrm{c}_4$ = 109
Hodge diamond
1
00
0 2 0
0 0 0 0
0 10 83 10 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 54 0 54 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 54 0 0 39 0 15
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 10t + 19t^{2} + 10t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(716)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(711)

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( 5t + 1 \right) ^{4} \left( 385637t^{5} + 177932t^{4} + 25896t^{3} + 1312t^{2} + 28t - 1 \right) \left( 180340658655268t^{10} + 232585725958436t^{9} + 129208156870786t^{8} + 40696201061053t^{7} + 8092890727315t^{6} + 1076684207269t^{5} + 98343778355t^{4} + 5910710895t^{3} + 187003365t^{2} + 979979t + 943 \right)D^{6} + 2 \left( 5t + 1 \right) ^{3} \left( 3651166605546683250090t^{16} + 6575091408629791143604t^{15} + 5346954627594044507561t^{14} + 2602403670567859890490t^{13} + 848107064160642481375t^{12} + 196439196725117080253t^{11} + 33515842231737383787t^{10} + 4297750184152324182t^{9} + 415522074585954655t^{8} + 29689847566889450t^{7} + 1490125957590705t^{6} + 48232658956260t^{5} + 929572604525t^{4} + 14924779465t^{3} + 367928759t^{2} + 1470440t + 943 \right)D^{5} + \left( 5t + 1 \right) ^{2} \left( 304263883795556937507500t^{17} + 572197141775802095762900t^{16} + 488596207642241219699666t^{15} + 251281983218055733387183t^{14} + 87155306776777500997557t^{13} + 21656409167818549051754t^{12} + 3998842034974231365934t^{11} + 560571890415453397482t^{10} + 60028194734373471161t^{9} + 4842213257203941645t^{8} + 282344889712995702t^{7} + 10995138635278715t^{6} + 247339263918079t^{5} + 2255374455924t^{4} - 11616286246t^{3} - 632270628t^{2} - 2184553t - 943 \right)D^{4} + 2 t \left( 5t + 1 \right) \left( 3194770779853347843828750t^{17} + 6345239812571308506205000t^{16} + 5754320488518574498386825t^{15} + 3162647344415454721024262t^{14} + 1180235133451560871226768t^{13} + 317805005422370800959852t^{12} + 64075822017326748463540t^{11} + 9892478515730197317735t^{10} + 1180305579353730699832t^{9} + 108096299295777360266t^{8} + 7393695749227291775t^{7} + 358035526975214646t^{6} + 11230403735530650t^{5} + 197710910070774t^{4} + 1607943074316t^{3} + 7309821381t^{2} + 8102472t + 4324 \right)D^{3} + 4 t ^{2} \left( 17647305260142302375435000t^{17} + 37303959089611932590728000t^{16} + 36198989624286267798775650t^{15} + 21412740381905965237933525t^{14} + 8653222866860730996140624t^{13} + 2539276617224480826893378t^{12} + 561593841782983118257476t^{11} + 95803583422739747807593t^{10} + 12754162910863336897511t^{9} + 1323542128473766052542t^{8} + 105254988735511406242t^{7} + 6183471261330256269t^{6} + 252294808580020070t^{5} + 6488939236776836t^{4} + 91138850291264t^{3} + 644504142557t^{2} + 3697522531t + 5772308 \right)D^{2} + 8 t ^{2} \left( 9584312339560043531486250t^{17} + 19752303312264012931447250t^{16} + 18653653807838321971983025t^{15} + 10720682098200714617956515t^{14} + 4203364352701222022561520t^{13} + 1195461024747302015888502t^{12} + 256043745001524063776256t^{11} + 42259763736105505551353t^{10} + 5431441761727402788919t^{9} + 541481523955221354546t^{8} + 40982138555335491641t^{7} + 2256138906343884973t^{6} + 84249747410379678t^{5} + 1915483354284018t^{4} + 22841780974902t^{3} + 145830229095t^{2} + 754794737t + 1293796 \right)D + 48 t ^{2} \left( 651994036704764866087500t^{17} + 1320365072877786666017500t^{16} + 1223481337141445325744950t^{15} + 688977310943687332957770t^{14} + 264358249530972825950983t^{13} + 73504800992488558612847t^{12} + 15379067758194797567685t^{11} + 2477061345534499606239t^{10} + 310024510272366159093t^{9} + 29961313921313674188t^{8} + 2179814468953371918t^{7} + 113785564027229694t^{6} + 3945231433010391t^{5} + 80754549930365t^{4} + 840450470935t^{3} + 4964362017t^{2} + 17622209t + 26404 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 58.7467 λ ≈ 58.7467; multiplicity 1; Jordan blocks 1 λ ≈ -10.36276 - 12.41289i; multiplicity 1; Jordan blocks 1 λ ≈ -10.36276 + 12.41289i; multiplicity 1; Jordan blocks 1 λ ≈ -5.010592 - 0.007707024i; multiplicity 1; Jordan blocks 1 λ ≈ -5.010592 + 0.007707024i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 1, 3 4
Sector dimension
9
Spectral radius
$58.7466959$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$58.7467$11
$-10.36276 - 12.41289\mathrm{i}$11
$-10.36276 + 12.41289\mathrm{i}$11
$-5.010592 - 0.007707024\mathrm{i}$11
$-5.010592 + 0.007707024\mathrm{i}$11
$-5$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{4} \left(\lambda^{5} - 28 \lambda^{4} - 1312 \lambda^{3} - 25896 \lambda^{2} - 177932 \lambda - 385637\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{5} - 28 \lambda^{4} - 1312 \lambda^{3} - 25896 \lambda^{2} - 177932 \lambda - 385637\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 64 & 104 & 2052 & 840 & 2418 & 40528 & 21248 & 219860 \\ 1 & -1 & 12 & 136 & 64 & 176 & 3456 & 1908 & 19280 \\ 1 & 4 & 4 & 192 & 96 & 200 & 4020 & 2112 & 21248 \\ 0 & 1 & 1 & 3 & 0 & 16 & 168 & 96 & 1212 \\ 0 & 1 & 0 & 12 & -1 & 13 & 168 & 80 & 996 \\ 0 & 0 & 1 & 4 & 4 & 0 & 160 & 96 & 840 \\ 0 & 0 & 0 & 1 & 0 & 2 & 3 & 4 & 64 \\ 0 & 0 & 0 & 1 & 1 & -\frac{3}{4} & 13 & 0 & 40 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0\end{pmatrix}\notag\end{equation}$$