Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-80-23-b

·

This is a Fano fourfold of K3 type, appearing as case K3-31 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ3P3 × ℙ6P6
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 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
E^(2w4) + E^(w1 + w4) + E^(w2 + w4)
Bundle
O_2(2) + O_1(1) box O_2(1) + Q_1^v(1) box O_2(1)
ZL62 label
30600.Cw9w2
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;7) \\ \det\mathcal{E} &\cong \mathcal{O}(3;6) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 23
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 80
$\chi(X, \mathrm{T}_X)$ = -27

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 80
$\mathrm{c}_1^2\mathrm{c}_2$ = 104
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 12
$\mathrm{c}_4$ = 36
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 28 1 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 + 27 0 x0 + 27 x2
$\mathrm{HH}^{ 3 }$ = x1 + 5 0 0 x1 5
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 13 0 0 x1 - x2 - 10 0 23
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 18t + 42t^{2} + 18t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(628)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) No match

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) ^{2} \left( 3t + 1 \right) ^{3} \left( 8864t^{4} + 4144t^{3} + 336t^{2} + 8t - 1 \right) \left( 1324971822096t^{10} + 2900048806920t^{9} + 2724054821300t^{8} + 1422640123648t^{7} + 452899381360t^{6} + 93116687370t^{5} + 13393092587t^{4} + 1434080440t^{3} + 90763136t^{2} + 527168t + 2240 \right)D^{6} + \left( 2t + 1 \right) \left( 3t + 1 \right) ^{2} \left( 1479813329113426944t^{16} + 4693152660533959680t^{15} + 6695886330632403456t^{14} + 5663474718853358848t^{13} + 3157673429335936096t^{12} + 1225852788544482416t^{11} + 343575332903360696t^{10} + 71794494110665668t^{9} + 11519039424801464t^{8} + 1425980580817848t^{7} + 128508146414434t^{6} + 7326111556071t^{5} + 222607695826t^{4} + 6276453912t^{3} + 357773184t^{2} + 1543424t + 4480 \right)D^{5} + \left( 3t + 1 \right) \left( 73990666455671347200t^{18} + 277484069279932554240t^{17} + 474896622058133183424t^{16} + 490372617650287177824t^{15} + 340719552762535788048t^{14} + 168542531423560990064t^{13} + 61448134046091137728t^{12} + 16943221622665183440t^{11} + 3614563402537140864t^{10} + 607057663851286116t^{9} + 80050168561981584t^{8} + 7923116850666034t^{7} + 529603275013919t^{6} + 19621503582272t^{5} + 245014184113t^{4} - 3596985960t^{3} - 328036832t^{2} - 1345984t - 2240 \right)D^{4} + t \left( 932282397341458974720t^{18} + 3620618792688703472640t^{17} + 6457099841496614229120t^{16} + 6997761239866352178624t^{15} + 5145077295302347745376t^{14} + 2718407767143543062656t^{13} + 1069339810711023851328t^{12} + 321344188373241806752t^{11} + 75384620497665855732t^{10} + 14043344983829003168t^{9} + 2084931288003253256t^{8} + 240569932896082108t^{7} + 20188820655551917t^{6} + 1096543484492004t^{5} + 31818561185073t^{4} + 374965676164t^{3} + 3367062280t^{2} + 19181120t + 7168 \right)D^{3} + 2 t ^{2} \left( 1029950077062945153024t^{17} + 3842334441994056786432t^{16} + 6574117684987115631072t^{15} + 6823420357482654041808t^{14} + 4795923637037738507448t^{13} + 2418662903609119834816t^{12} + 907749748980755369032t^{11} + 260640350048151208116t^{10} + 58621460951075349162t^{9} + 10500840721399215696t^{8} + 1494087263041691858t^{7} + 162644066074609124t^{6} + 12490443294838706t^{5} + 592531961483888t^{4} + 14116431783147t^{3} + 150348784874t^{2} + 1835809904t + 7723120 \right)D^{2} + 8 t ^{2} \left( 279684719202437692416t^{17} + 1012032856246050554112t^{16} + 1677401534540218073040t^{15} + 1683816788887221354840t^{14} + 1142638699555756268148t^{13} + 555597385367452512720t^{12} + 200977650903942684720t^{11} + 55693478605927022118t^{10} + 12122963622153117771t^{9} + 2104735178520276724t^{8} + 288748108695128676t^{7} + 29820238746645550t^{6} + 2111285729084439t^{5} + 88416316979856t^{4} + 1761951845280t^{3} + 18518982224t^{2} + 216095232t + 730240 \right)D + 16 t ^{2} \left( 57078514122946467840t^{17} + 202213540250764807680t^{16} + 327757681033802447328t^{15} + 321269064664781860848t^{14} + 212549393372207303736t^{13} + 100630022986664130240t^{12} + 35427147897558319164t^{11} + 9563489749103172588t^{10} + 2031828829365729798t^{9} + 344432847382374636t^{8} + 45879719299460169t^{7} + 4533033416380242t^{6} + 299407357376196t^{5} + 11254717040336t^{4} + 193228599912t^{3} + 2126676784t^{2} + 19544448t + 49280 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × ℙ6P6

|λ| = 26.78332 λ ≈ 26.78332; multiplicity 1; Jordan blocks 1 λ ≈ -8.058252 - 7.691851i; multiplicity 1; Jordan blocks 1 λ ≈ -8.058252 + 7.691851i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 3; Jordan blocks 1, 2 λ ≈ -2.666819; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 3 2
Sector dimension
9
Spectral radius
$26.7833228$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$26.78332$11
$-8.058252 - 7.691851\mathrm{i}$11
$-8.058252 + 7.691851\mathrm{i}$11
$-3$31, 2
$-2.666819$11
$-2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{3} \left(\lambda^{4} - 8 \lambda^{3} - 336 \lambda^{2} - 4144 \lambda - 8864\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda^{4} - 8 \lambda^{3} - 336 \lambda^{2} - 4144 \lambda - 8864\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 20 & 24 & 264 & 126 & 312 & 2208 & 1296 & 8080 \\ 1 & -1 & 6 & 20 & 12 & 20 & 348 & 216 & 1152 \\ 1 & 2 & 0 & 48 & 28 & 44 & 420 & 240 & 1344 \\ 0 & 1 & 1 & -1 & -1 & 6 & 16 & 12 & 144 \\ 0 & 1 & 0 & 6 & 0 & 8 & 28 & 16 & 144 \\ 0 & 0 & 1 & 2 & 2 & -2 & 44 & 28 & 120 \\ 0 & 0 & 0 & 1 & -\frac{1}{5} & 3 & -\frac{3}{5} & \frac{9}{5} & \frac{144}{5} \\ 0 & 0 & 0 & 1 & \frac{8}{5} & -2 & \frac{34}{5} & -\frac{2}{5} & -\frac{32}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{4} & 1 & 0\end{pmatrix}\notag\end{equation}$$