Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-94-26-a

·

This is a Fano fourfold of K3 type, appearing as case GM-20 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ3P3 × ℙ8P8
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_9node 10: fundamental weight ω_10node 11: fundamental weight ω_11ω1ω2ω3ω4ω5ω6ω7ω8ω9ω10ω11
Bundle
E^(2w4) + E^(w2 + w4) + E^(w3 + w4)
Bundle
O_2(2) + Q_1^v(1) box O_2(1) + Q_1 box O_2(1)
ZL62 label
30800.EwJwB
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;9) \\ \det\mathcal{E} &\cong \mathcal{O}(3;8) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 26
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 94
$\chi(X, \mathrm{T}_X)$ = -28

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 94
$\mathrm{c}_1^2\mathrm{c}_2$ = 112
$\mathrm{c}_2^2$ = 130
$\mathrm{c}_1\mathrm{c}_3$ = 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 + 28 0 x0 + 28 x2
$\mathrm{HH}^{ 3 }$ = x1 + 10 0 0 x1 10
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 17 0 0 x1 - x2 - 9 0 26
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 21t + 50t^{2} + 21t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(566)
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) ^{2} \left( 22928t^{5} + 17040t^{4} + 3848t^{3} + 192t^{2} + 5t - 1 \right) \left( 23621295652352t^{13} + 79375097748736t^{12} + 125907268076032t^{11} + 123675167362304t^{10} + 81399657537440t^{9} + 36589184671568t^{8} + 11207839011648t^{7} + 2317188825464t^{6} + 321780008724t^{5} + 30779386720t^{4} + 2144603639t^{3} + 94715679t^{2} + 583464t + 2538 \right)D^{6} + \left( 2t + 1 \right) \left( 3t + 1 \right) \left( 68240222406357958656t^{20} + 316487406720211705856t^{19} + 699758636031575965696t^{18} + 976545860463596376064t^{17} + 954038834168534585344t^{16} + 683605815330842726400t^{15} + 367318993557811787264t^{14} + 149558673883225583872t^{13} + 46346610461000822848t^{12} + 10948242968578016608t^{11} + 1974883391528059648t^{10} + 273520445103867696t^{9} + 29415216832535400t^{8} + 2478095948186852t^{7} + 160400934221030t^{6} + 7499773142103t^{5} + 255983967044t^{4} + 9607742743t^{3} + 370538520t^{2} + 1699632t + 5076 \right)D^{5} + \left( 3412011120317897932800t^{22} + 17851542764612587192320t^{21} + 44668830673217009623040t^{20} + 70894452825934724349952t^{19} + 79452354060103171100672t^{18} + 66170174674634944798720t^{17} + 42047593396993939688448t^{16} + 20676058318705774887936t^{15} + 7926559167040626170240t^{14} + 2378028739628482606464t^{13} + 559387975694449474464t^{12} + 103377964863979837632t^{11} + 15062464369532679584t^{10} + 1737489201582617968t^{9} + 158009558924331160t^{8} + 10938645635240156t^{7} + 520810925726847t^{6} + 13283760380260t^{5} + 22988376326t^{4} - 7746958796t^{3} - 333727131t^{2} - 1399020t - 2538 \right)D^{4} + t \left( 14330446705335171317760t^{21} + 72283118044885253160960t^{20} + 174733924951766494699520t^{19} + 268224883351314073722880t^{18} + 290469518100340347523072t^{17} + 233035690131150315096064t^{16} + 141995369724396766382080t^{15} + 66600967185929843173888t^{14} + 24227331698523984953984t^{13} + 6866081493991305517952t^{12} + 1521457088545791526304t^{11} + 264863782695376896640t^{10} + 36481870322138702480t^{9} + 4000373745655392224t^{8} + 346346223285213448t^{7} + 22606710355477940t^{6} + 999461613095211t^{5} + 24776254298711t^{4} + 270196333381t^{3} + 2182713393t^{2} + 12431982t + 6210 \right)D^{3} + 2 t ^{2} \left( 15831731598275046408192t^{20} + 77579103373151559131136t^{19} + 182511131286617084878848t^{18} + 272889289686325904742400t^{17} + 287517801902166752413696t^{16} + 223735939562511829956608t^{15} + 131670685276640158593536t^{14} + 59369837517613126704896t^{13} + 20669617209347597290112t^{12} + 5586397636808984736160t^{11} + 1178319615569002770560t^{10} + 195381732891780819928t^{9} + 25709882091148387328t^{8} + 2698353422596249700t^{7} + 221979903961822448t^{6} + 13433957412455534t^{5} + 527187685414315t^{4} + 11028893841175t^{3} + 115607617071t^{2} + 1273361010t + 5219961 \right)D^{2} + 8 t ^{2} \left( 4299134011600551395328t^{20} + 20613712063158002577408t^{19} + 47514982917714988425216t^{18} + 69644653682454121671680t^{17} + 71846440427108524014592t^{16} + 54589056949933138317056t^{15} + 31251472892788185555968t^{14} + 13652689450124326806208t^{13} + 4588085569621608680192t^{12} + 1193484351478905131648t^{11} + 241960840340677862464t^{10} + 38593742751212976936t^{9} + 4896554123811095664t^{8} + 495385755515528210t^{7} + 38887215716056172t^{6} + 2188315945840802t^{5} + 76485756195468t^{4} + 1360653269075t^{3} + 14257391301t^{2} + 153725898t + 512676 \right)D + 32 t ^{2} \left( 438687144040872591360t^{20} + 2072209004978001039360t^{19} + 4709457851233825013760t^{18} + 6807678065553554847744t^{17} + 6918697079616800278528t^{16} + 5167081425120004871168t^{15} + 2898874051246851859968t^{14} + 1237058234052495900032t^{13} + 404844082133462104384t^{12} + 102308074075806313840t^{11} + 20126760461320937248t^{10} + 3116966360712593456t^{9} + 384528790308194008t^{8} + 37770111522481376t^{7} + 2848310875917100t^{6} + 150346696454532t^{5} + 4740938026190t^{4} + 73337890673t^{3} + 786356523t^{2} + 6939054t + 17766 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 5)

|λ| = 22.59713 λ ≈ 22.59713; multiplicity 1; Jordan blocks 1 λ ≈ -6.14285 - 9.747408i; multiplicity 1; Jordan blocks 1 λ ≈ -6.14285 + 9.747408i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 2 λ ≈ -2.655716 - 0.7685229i; multiplicity 1; Jordan blocks 1 λ ≈ -2.655716 + 0.7685229i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
9
Spectral radius
$22.5971308$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22.59713$11
$-6.14285 - 9.747408\mathrm{i}$11
$-6.14285 + 9.747408\mathrm{i}$11
$-3$22
$-2.655716 - 0.7685229\mathrm{i}$11
$-2.655716 + 0.7685229\mathrm{i}$11
$-2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda^{5} - 5 \lambda^{4} - 192 \lambda^{3} - 3848 \lambda^{2} - 17040 \lambda - 22928\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right)^{2} \left(\lambda^{5} - 5 \lambda^{4} - 192 \lambda^{3} - 3848 \lambda^{2} - 17040 \lambda - 22928\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 16 & 96 & 84 & 156 & 512 & 672 & 880 \\ 1 & -1 & 6 & 4 & 0 & 8 & 72 & 72 & 160 \\ 1 & 1 & 0 & 12 & 20 & 20 & 132 & 168 & 176 \\ 0 & 1 & 2 & 0 & 0 & 4 & 8 & 8 & 36 \\ 0 & 2 & 1 & 6 & -1 & 10 & 4 & 8 & 36 \\ 0 & 0 & 1 & 0 & 1 & -2 & 16 & 16 & 24 \\ 0 & 0 & 0 & 1 & 3 & 1 & 5 & 12 & 4 \\ 0 & 0 & 0 & 1 & -1 & 2 & -2 & -6 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$