Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-61-20-a

·

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

Defining data
Ambient
ℙ4P4 × ℙ5P5
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_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w4 + w5) + E^(3w1 + w5)
Bundle
Q_1 box O_2(1) + O_1(3) box O_2(1)
ZL62 label
4050.wLx33E
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;6) \\ \det\mathcal{E} &\cong \mathcal{O}(4;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 20
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 61
$\chi(X, \mathrm{T}_X)$ = -74

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 61
$\mathrm{c}_1^2\mathrm{c}_2$ = 106
$\mathrm{c}_2^2$ = 184
$\mathrm{c}_1\mathrm{c}_3$ = -46
$\mathrm{c}_4$ = 149
Hodge diamond
1
00
0 2 0
0 0 0 0
0 15 113 15 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 + 74 0 x0 + 74 x1
$\mathrm{HH}^{ 3 }$ = x1 + 10 0 0 x1 10
$\mathrm{HH}^{ 4 }$ = 84 0 0 64 0 20
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 15t + 29t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(684)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(683)

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( t + 1 \right) ^{3} \left( 30773t^{5} + 62796t^{4} + 33198t^{3} + 1132t^{2} - 27t - 1 \right) \left( 1405236007946t^{9} + 5062919472248t^{8} + 7063402942414t^{7} + 3870625013745t^{6} + 361330282570t^{5} + 12887875831t^{4} + 2785867214t^{3} + 240837875t^{2} + 17808t + 2989 \right)D^{6} + 2 \left( t + 1 \right) ^{2} \left( 454054940561483709t^{15} + 2658475607658019541t^{14} + 6663889724134512487t^{13} + 8999406914552114939t^{12} + 6789624796137179450t^{11} + 2705595644409774526t^{10} + 495933301171105680t^{9} + 43376501876443907t^{8} + 3860916832492820t^{7} + 560127849917328t^{6} + 40069781296101t^{5} + 578000614137t^{4} + 6575183913t^{3} + 490940929t^{2} - 68936t + 2989 \right)D^{5} + \left( t + 1 \right) \left( 7567582342691395150t^{16} + 47101565719853804460t^{15} + 127305410972359950792t^{14} + 189644691801832191471t^{13} + 164292935415002413684t^{12} + 81034449629403123334t^{11} + 21050833965064843714t^{10} + 2629628441914195541t^{9} + 218293851075412347t^{8} + 28282515978013698t^{7} + 3692648362571802t^{6} + 204234368438067t^{5} + 1505472577312t^{4} + 10240359268t^{3} - 914919956t^{2} - 585039t - 2989 \right)D^{4} + 2 t \left( 15891922919651929815t^{16} + 106928204293067357165t^{15} + 316432957068101625790t^{14} + 527229700343489277144t^{13} + 529227153387883571999t^{12} + 320542554959081242916t^{11} + 112238787498630503653t^{10} + 20995338070914750193t^{9} + 2081562615937649862t^{8} + 206506574441696839t^{7} + 33203638870937422t^{6} + 3447419007899344t^{5} + 138858997677041t^{4} + 477109237464t^{3} + 4871656319t^{2} + 34540247t + 2499 \right)D^{3} + 4 t ^{2} \left( 17556791035044036748t^{15} + 111472260560463617164t^{14} + 310179069981149983237t^{13} + 480276639715860051684t^{12} + 439290748440321225372t^{11} + 235939531377685065809t^{10} + 70789506554338463638t^{9} + 11021118451831941442t^{8} + 997095283935776208t^{7} + 121990918963888860t^{6} + 19992387607621539t^{5} + 1698494307434476t^{4} + 49083739573852t^{3} + 75260776435t^{2} + 2975430254t + 11808258 \right)D^{2} + 8 t ^{2} \left( 9535153751791157889t^{15} + 57891121840846899059t^{14} + 153523552091706328833t^{13} + 224027197203397153665t^{12} + 189530886211180009476t^{11} + 91790104877777884217t^{10} + 24079557923770761719t^{9} + 3227557916066876108t^{8} + 291572138420693795t^{7} + 43024857701010929t^{6} + 6530530471407735t^{5} + 454990510646491t^{4} + 8856516970708t^{3} + 11677643975t^{2} + 772312405t + 2061556 \right)D + 96 t ^{2} \left( 324324957543916935t^{15} + 1908193641476876885t^{14} + 4889136026576097093t^{13} + 6826438517278243755t^{12} + 5435668654507795738t^{11} + 2421536110150274699t^{10} + 568263123047146763t^{9} + 67729195683019758t^{8} + 6424002710861187t^{7} + 1078313861459019t^{6} + 148595912809403t^{5} + 8698125095847t^{4} + 108521234532t^{3} + 310794925t^{2} + 12317165t + 23912 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × ℙ5P5

|λ| = 34.70483 λ ≈ 34.70483; multiplicity 1; Jordan blocks 1 λ ≈ -31.27624; multiplicity 1; Jordan blocks 1 λ ≈ -28.43202; multiplicity 1; Jordan blocks 1 λ ≈ -1; multiplicity 3; Jordan blocks 3 λ ≈ -0.9982852 - 0.02388165i; multiplicity 1; Jordan blocks 1 λ ≈ -0.9982852 + 0.02388165i; multiplicity 1; Jordan blocks 1 3
Sector dimension
8
Spectral radius
$34.7048288$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$34.70483$11
$-31.27624$11
$-28.43202$11
$-1$33
$-0.9982852 - 0.02388165\mathrm{i}$11
$-0.9982852 + 0.02388165\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} + 27 \lambda^{4} - 1132 \lambda^{3} - 33198 \lambda^{2} - 62796 \lambda - 30773\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 1\right)^{3} \left(\lambda^{5} + 27 \lambda^{4} - 1132 \lambda^{3} - 33198 \lambda^{2} - 62796 \lambda - 30773\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 48 & 48 & 216 & 108 & 5712 & 3456 & 8660 \\ 1 & -6 & 4 & 120 & 228 & 228 & -798 & 3008 \\ 1 & 6 & -1 & 88 & -56 & 312 & 1167 & 2704 \\ 0 & 1 & 2 & -1 & 15 & 160 & -38 & 72 \\ 0 & 1 & 0 & 4 & -15 & 48 & 156 & 144 \\ 0 & 0 & 0 & 2 & 1 & -1 & 6 & 48 \\ 0 & 0 & 0 & 0 & 1 & 4 & -6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & \frac{5}{4} & 0\end{pmatrix}\notag\end{equation}$$