Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-68-21-b

·

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

Defining data
Ambient
ℙ2P2 × ℙ8P8
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_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9node 10: fundamental weight ω_10ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10
Bundle
2 copies of E^(2w3) + 2 copies of E^(w2 + w3)
Bundle
2 copies of O_2(2) + 2 copies of Q_1 box O_2(1)
ZL62 label
20800.CCw9w9
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;9) \\ \det\mathcal{E} &\cong \mathcal{O}(2;8) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 21
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 68
$\chi(X, \mathrm{T}_X)$ = -42

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 68
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 148
$\mathrm{c}_1\mathrm{c}_3$ = -8
$\mathrm{c}_4$ = 64
Hodge diamond
1
00
0 2 0
0 0 0 0
0 4 50 4 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 + 42 0 x0 + 42 x2
$\mathrm{HH}^{ 3 }$ = x1 + 6 0 0 x1 6
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 27 0 0 x1 - x2 + 6 0 21
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 16t + 34t^{2} + 16t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(654)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(661)

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( 3t + 1 \right) \left( 4t + 1 \right) ^{4} \left( 21248t^{4} + 6144t^{3} - 160t^{2} + 32t - 1 \right) \left( 729374261248t^{10} + 1020981805056t^{9} + 604167929856t^{8} + 212923613184t^{7} + 51476533248t^{6} + 8516828160t^{5} + 872911872t^{4} + 51147392t^{3} + 1899160t^{2} + 5970t + 17 \right)D^{6} + 2 \left( 4t + 1 \right) ^{3} \left( 1952715782177685504t^{16} + 4033570758956417024t^{15} + 3688168094238244864t^{14} + 2017394610640257024t^{13} + 753137488220913664t^{12} + 204808679071416320t^{11} + 41149315593338880t^{10} + 5975275123507200t^{9} + 608043050323968t^{8} + 42791108507648t^{7} + 2063029146368t^{6} + 64930701056t^{5} + 2395173968t^{4} + 98016136t^{3} + 3777011t^{2} + 9074t + 17 \right)D^{5} + \left( 4t + 1 \right) ^{2} \left( 130181052145179033600t^{17} + 279925945578133913600t^{16} + 269475708625439686656t^{15} + 157140504016981065728t^{14} + 63262501183269371904t^{13} + 18756880761287081984t^{12} + 4173591322444693504t^{11} + 688573021792108544t^{10} + 82285135445262336t^{9} + 6986888224579584t^{8} + 412879267661824t^{7} + 15471830784512t^{6} + 234359857792t^{5} - 4935649376t^{4} - 249058264t^{3} - 6301710t^{2} - 12895t - 17 \right)D^{4} + 2 t \left( 4t + 1 \right) \left( 1093520838019503882240t^{17} + 2482699523196515778560t^{16} + 2547840164735295160320t^{15} + 1598171803149441957888t^{14} + 697251548423335706624t^{13} + 225758014629872664576t^{12} + 55537060552330182656t^{11} + 10335833085121134592t^{10} + 1430842291532726272t^{9} + 144809704500166656t^{8} + 10549704024293376t^{7} + 534112524042240t^{6} + 16717437872128t^{5} + 241401984256t^{4} + 936316928t^{3} - 10273568t^{2} + 29929t + 44 \right)D^{3} + 4 t ^{2} \left( 4832320655629045727232t^{17} + 11698954120190310219776t^{16} + 12904543836103268892672t^{15} + 8755847972681014050816t^{14} + 4151489172270775009280t^{13} + 1468529011472501571584t^{12} + 398492189206153003008t^{11} + 83149789477551996928t^{10} + 13193788241765990400t^{9} + 1568433185984610304t^{8} + 137643477581987840t^{7} + 8722690417213440t^{6} + 375884569591808t^{5} + 9342109905408t^{4} + 88854011776t^{3} + 301304192t^{2} + 5203488t + 23715 \right)D^{2} + 32 t ^{2} \left( 656112502811702329344t^{17} + 1540646657330744655872t^{16} + 1646344882482371887104t^{15} + 1084504388254335762432t^{14} + 501107069171645022208t^{13} + 173171659962239156224t^{12} + 45846291047526694912t^{11} + 9284834464817217536t^{10} + 1419909998378287104t^{9} + 161614753787838464t^{8} + 13501753292365824t^{7} + 806631233235968t^{6} + 31863897592576t^{5} + 668698246912t^{4} + 3607496256t^{3} - 2065948t^{2} + 435616t + 1513 \right)D + 64 t ^{2} \left( 133900510777898434560t^{17} + 307828694948543201280t^{16} + 321689588382088298496t^{15} + 207541986952659075072t^{14} + 94174980181856354304t^{13} + 32007638235551367168t^{12} + 8320272807622606848t^{11} + 1646911769771769856t^{10} + 244738937655132160t^{9} + 26925768837922816t^{8} + 2163801549742080t^{7} + 123233543918592t^{6} + 4522681060608t^{5} + 81799747648t^{4} + 199434992t^{3} - 2682272t^{2} + 29276t + 119 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ8P8

|λ| = 33.3091 λ ≈ 33.3091; multiplicity 1; Jordan blocks 1 λ ≈ 0.8720433 - 14.42809i; multiplicity 1; Jordan blocks 1 λ ≈ 0.8720433 + 14.42809i; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 λ ≈ -3.053191; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 4
Sector dimension
9
Spectral radius
$33.3091046$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$33.3091$11
$0.8720433 - 14.42809\mathrm{i}$11
$0.8720433 + 14.42809\mathrm{i}$11
$-4$41, 3
$-3.053191$11
$-3$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{4} \left(\lambda^{4} - 32 \lambda^{3} + 160 \lambda^{2} - 6144 \lambda - 21248\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 4\right)^{3} \left(\lambda^{4} - 32 \lambda^{3} + 160 \lambda^{2} - 6144 \lambda - 21248\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 8 & 48 & 336 & 168 & 720 & 6432 & 2720 & 33920 \\ 1 & 0 & 8 & 24 & -8 & 88 & 192 & 48 & 1984 \\ 1 & 0 & 4 & 32 & 24 & 80 & 912 & 432 & 5440 \\ 0 & 1 & 1 & 4 & -2 & 16 & 8 & -16 & 0 \\ 0 & 1 & 0 & 8 & 1 & 12 & 112 & 40 & 432 \\ 0 & 0 & 1 & 0 & 1 & 0 & 40 & 24 & 336 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 16 \\ 0 & 0 & 0 & 1 & 1 & -1 & 16 & 4 & 64 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$