Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-64-20-b

·

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

Defining data
Ambient
ℙ2P2 × ℙ7P7
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_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
2 copies of E^(2w3) + E^(w1 + w3) + E^(w2 + w3)
Bundle
2 copies of O_2(2) + O_1(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
20700.BBw8w1
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;8) \\ \det\mathcal{E} &\cong \mathcal{O}(2;7) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 20
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 64
$\chi(X, \mathrm{T}_X)$ = -38

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 64
$\mathrm{c}_1^2\mathrm{c}_2$ = 100
$\mathrm{c}_2^2$ = 148
$\mathrm{c}_1\mathrm{c}_3$ = -4
$\mathrm{c}_4$ = 56
Hodge diamond
1
00
0 2 0
0 0 0 0
0 3 44 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 + 38 0 x0 + 38 x2
$\mathrm{HH}^{ 3 }$ = x1 + 4 0 0 x1 4
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 21 0 0 x1 - x2 + 1 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 15t + 32t^{2} + 15t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(670)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(671)

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) ^{2} \left( 4t + 1 \right) ^{4} \left( 5296t^{3} - 120t^{2} + 35t - 1 \right) \left( 22520448794112t^{10} + 33131216228736t^{9} + 20209640074784t^{8} + 6571817032584t^{7} + 1229788714304t^{6} + 135834145222t^{5} + 9236765478t^{4} + 448884168t^{3} + 15592806t^{2} + 43109t + 72 \right)D^{6} + \left( 5t + 1 \right) \left( 4t + 1 \right) ^{3} \left( 50092684661719203840t^{15} + 87257534300902785024t^{14} + 65578001927520352256t^{13} + 27822098779146121216t^{12} + 7362534020636560256t^{11} + 1277984142129105920t^{10} + 151009847326464328t^{9} + 12695895913728464t^{8} + 802020735757714t^{7} + 39831702686504t^{6} + 1639423593890t^{5} + 58109472692t^{4} + 1485940259t^{3} + 61275954t^{2} + 129039t + 144 \right)D^{5} + \left( 4t + 1 \right) ^{2} \left( 8348780776953200640000t^{17} + 17254064391839448268800t^{16} + 15802945893783622787072t^{15} + 8445274864232085272576t^{14} + 2924696772362116121088t^{13} + 691336312035924417664t^{12} + 114960207759757176800t^{11} + 13765429312932684456t^{10} + 1218847520138701304t^{9} + 82084765580330950t^{8} + 4175388940986424t^{7} + 141858545293808t^{6} + 2030816086716t^{5} - 36293783043t^{4} - 1964119446t^{3} - 49581916t^{2} - 89431t - 72 \right)D^{4} + t \left( 4t + 1 \right) \left( 140259517052813770752000t^{17} + 311128477741298751897600t^{16} + 308348647281029652152320t^{15} + 180058828934342136381440t^{14} + 68918050974933447856128t^{13} + 18237024233143706317824t^{12} + 3440435942074549883392t^{11} + 472879202689746973376t^{10} + 48402363060093811160t^{9} + 3785602731907672424t^{8} + 229680941969768454t^{7} + 10417040123536106t^{6} + 309605231714982t^{5} + 4593177477378t^{4} + 12545818531t^{3} - 254320825t^{2} + 129180t + 405 \right)D^{3} + t ^{2} \left( 1239626969762011231027200t^{17} + 2966343906226331115847680t^{16} + 3196117307890214708707328t^{15} + 2047852564851273890004992t^{14} + 869230393711093877719040t^{13} + 258119694853356552241152t^{12} + 55330601124393698915328t^{11} + 8741764227484388463616t^{10} + 1036521195061329728704t^{9} + 94106421021657324480t^{8} + 6656304269610873648t^{7} + 363042906423712432t^{6} + 14138788248607436t^{5} + 328152125657292t^{4} + 3003442138738t^{3} - 2712833322t^{2} + 109449703t + 458928 \right)D^{2} + 32 t ^{2} \left( 42077855115844131225600t^{17} + 98367508059665888378880t^{16} + 103223110918316150669312t^{15} + 64187651519633660465152t^{14} + 26343009834088953972736t^{13} + 7535877817432998274816t^{12} + 1551327263096323434560t^{11} + 234976418515985927248t^{10} + 26734599344826515668t^{9} + 2337115857899036968t^{8} + 159308326628302234t^{7} + 8254982456961466t^{6} + 292103385314730t^{5} + 5518946564776t^{4} + 22844705588t^{3} - 302274207t^{2} + 2832318t + 8280 \right)D + 96 t ^{2} \left( 5724878247053623296000t^{17} + 13169903952075640012800t^{16} + 13566703213361723015168t^{15} + 8258433332738900697088t^{14} + 3307821762476767934464t^{13} + 920705605026199883264t^{12} + 183934011761726740864t^{11} + 26997871970871656256t^{10} + 2978690774005534256t^{9} + 253132344330821088t^{8} + 16759893464838664t^{7} + 831420305702192t^{6} + 27054223073676t^{5} + 421365044676t^{4} - 66003774t^{3} - 38957922t^{2} + 114183t + 432 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ7P7

|λ| = 35.78264 λ ≈ 35.78264; multiplicity 1; Jordan blocks 1 λ ≈ -0.3913176 - 12.15943i; multiplicity 1; Jordan blocks 1 λ ≈ -0.3913176 + 12.15943i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 2; Jordan blocks 2 λ ≈ -4; multiplicity 4; Jordan blocks 1, 3 2 4
Sector dimension
9
Spectral radius
$35.7826352$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$35.78264$11
$-0.3913176 - 12.15943\mathrm{i}$11
$-0.3913176 + 12.15943\mathrm{i}$11
$-5$22
$-4$41, 3

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{4} \left(\lambda + 5\right)^{2} \left(\lambda^{3} - 35 \lambda^{2} + 120 \lambda - 5296\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{3} \left(\lambda + 5\right)^{2} \left(\lambda^{3} - 35 \lambda^{2} + 120 \lambda - 5296\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 56 & 456 & 168 & 816 & 7616 & 2912 & 40320 \\ 1 & -1 & 8 & 40 & 0 & 112 & 528 & 168 & 3584 \\ 1 & 1 & 4 & 48 & 24 & 88 & 1032 & 432 & 5824 \\ 0 & 1 & 1 & 3 & -1 & 16 & 40 & 0 & 240 \\ 0 & 1 & 0 & 8 & 0 & 8 & 112 & 40 & 480 \\ 0 & 0 & 1 & 1 & 1 & 0 & 48 & 24 & 336 \\ 0 & 0 & 0 & 1 & 0 & 3 & 1 & 1 & 32 \\ 0 & 0 & 0 & 1 & 1 & -2 & 14 & 2 & 48 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{1}{2} & 0\end{pmatrix}\notag\end{equation}$$