Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-82-24-b

·

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

Defining data
Ambient
ℙ4P4 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
E^(w1 + 2w5) + E^(w4 + w5) + E^(2w1)
Bundle
O_1(1) box O_2(2) + Q_1 box O_2(1) + O_1(2)
ZL62 label
40600.wOx331A
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;7) \\ \det\mathcal{E} &\cong \mathcal{O}(4;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 82
$\chi(X, \mathrm{T}_X)$ = -48

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 82
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 150
$\mathrm{c}_1\mathrm{c}_3$ = -12
$\mathrm{c}_4$ = 84
Hodge diamond
1
00
0 2 0
0 0 0 0
0 7 64 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 + x1 + 48 0 x0 + 48 x1
$\mathrm{HH}^{ 3 }$ = x1 + 14 0 0 x1 + 4 10
$\mathrm{HH}^{ 4 }$ = 49 0 0 25 0 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 42t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(645)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(653)

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) \left( 2t + 1 \right) ^{4} \left( 19616t^{4} + 7152t^{3} + 504t^{2} - 4t - 1 \right) \left( 93956761542784t^{11} - 158492251330176t^{10} - 429322315798120t^{9} - 326397268545476t^{8} - 118393811896072t^{7} - 22291403877312t^{6} - 1986581946188t^{5} - 19674090137t^{4} + 9021084168t^{3} + 101977026t^{2} - 4860888t - 1517 \right)D^{6} + 2 \left( 2t + 1 \right) ^{3} \left( 77408345045776539648t^{17} - 132378576645477036032t^{16} - 405363619995486305792t^{15} - 324616485998398895872t^{14} - 99269863568322154240t^{13} + 5782073838033088000t^{12} + 13496084177766408720t^{11} + 4362882192183252704t^{10} + 691037374685212148t^{9} + 51473489044028820t^{8} + 61366215607708t^{7} - 199648615779464t^{6} - 2450231300476t^{5} + 211582474627t^{4} - 15394693840t^{3} - 260003140t^{2} + 7270094t + 1517 \right)D^{5} + \left( 2t + 1 \right) ^{2} \left( 2580278168192551321600t^{18} - 3923129641868956303360t^{17} - 15589185187424652999168t^{16} - 16028432681960966872576t^{15} - 7815741821253970192096t^{14} - 1754433473162513061456t^{13} + 22533613856523315408t^{12} + 116853182596671342832t^{11} + 30556976261783159048t^{10} + 3570938213418562132t^{9} + 108719097967422772t^{8} - 19003387680491164t^{7} - 1821643685824900t^{6} - 36332518149980t^{5} + 302201389927t^{4} + 60461369174t^{3} + 384706874t^{2} - 10128998t - 1517 \right)D^{4} + 4 t \left( 2t + 1 \right) \left( 5418584153204357775360t^{18} - 6850351853664025108480t^{17} - 37085719521960885153280t^{16} - 47184906194892406755584t^{15} - 30231254089353580495200t^{14} - 10971928530010403788032t^{13} - 2109968253891394325128t^{12} - 81462897890171544208t^{11} + 58040755412901532648t^{10} + 13966369671403433128t^{9} + 1238408094281604212t^{8} - 7611367553558150t^{7} - 8557531396327918t^{6} - 304287828155294t^{5} + 12891605534081t^{4} + 713148906323t^{3} + 7148488415t^{2} - 4458531t + 925 \right)D^{3} + 2 t ^{2} \left( 47889962801653752528896t^{18} - 45532041595959850303488t^{17} - 362973126707262499933184t^{16} - 560800090085967595902976t^{15} - 443789834445696171555904t^{14} - 210915230869218429416480t^{13} - 62246706197537498441552t^{12} - 10686519635147267554072t^{11} - 667645154177843679432t^{10} + 120586635604981003920t^{9} + 30229312767972375324t^{8} + 2101390479931223798t^{7} - 71644972563810392t^{6} - 12955473258387126t^{5} + 171048635517058t^{4} + 45660027713383t^{3} + 1438142375954t^{2} + 7286300125t + 1633850 \right)D^{2} + 8 t ^{2} \left( 13004601967690458660864t^{18} - 14141953363307417436160t^{17} - 99398573234975763333632t^{16} - 148037543112945343254528t^{15} - 113888664761851944725344t^{14} - 52996343178049860834672t^{13} - 15542073613216461710504t^{12} - 2773337053447829471756t^{11} - 238865007980016265076t^{10} + 8761864755417198056t^{9} + 4230210820975208162t^{8} + 302995292861784229t^{7} - 8115807645768262t^{6} - 817885631882959t^{5} + 110212379168096t^{4} + 7695399148861t^{3} + 188217402846t^{2} + 453657565t + 628038 \right)D + 16 t ^{2} \left( 2654000401569481359360t^{18} - 3131209718363553792000t^{17} - 20387673910107543088128t^{16} - 29604324413234834850816t^{15} - 22306467762743059998528t^{14} - 10198968150027938176352t^{13} - 2958323898601706462144t^{12} - 532061582228061857184t^{11} - 50304279278047851512t^{10} + 61868160048894508t^{9} + 551243064038579188t^{8} + 42493531812244060t^{7} - 380169217550018t^{6} + 32311649092053t^{5} + 21468654617708t^{4} + 1025323363181t^{3} + 20371517894t^{2} + 20172622t + 39442 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ6P6

|λ| = 26.40714 λ ≈ 26.40714; multiplicity 1; Jordan blocks 1 λ ≈ -13.34364 - 4.651878i; multiplicity 1; Jordan blocks 1 λ ≈ -13.34364 + 4.651878i; multiplicity 1; Jordan blocks 1 λ ≈ -3.719863; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 4; Jordan blocks 1, 3 λ ≈ 2; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$26.4071416$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$26.40714$11
$-13.34364 - 4.651878\mathrm{i}$11
$-13.34364 + 4.651878\mathrm{i}$11
$-3.719863$11
$-2$41, 3
$2$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{4} \left(\lambda - 2\right) \left(\lambda^{4} + 4 \lambda^{3} - 504 \lambda^{2} - 7152 \lambda - 19616\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda - 2\right) \left(\lambda^{4} + 4 \lambda^{3} - 504 \lambda^{2} - 7152 \lambda - 19616\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 28 & 216 & 96 & 252 & 1536 & 1312 & 5040 \\ 1 & -2 & 5 & 32 & 36 & 30 & 360 & 276 & 672 \\ 1 & 2 & 0 & 56 & 28 & 60 & 144 & 108 & 864 \\ 0 & 1 & 1 & -2 & -2 & 3 & 60 & 60 & 144 \\ 0 & 1 & 0 & 5 & -2 & 6 & 6 & 8 & 108 \\ 0 & 0 & 1 & 2 & 4 & -2 & 24 & 12 & 0 \\ 0 & 0 & 0 & 1 & -\frac{4}{3} & 2 & -10 & -\frac{14}{3} & 16 \\ 0 & 0 & 0 & 1 & 3 & 0 & 15 & 8 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$