Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-129-33-b

·

This is a Fano fourfold of K3 type, appearing as case R-62 in Bernardara–Fatighenti–Manivel–Tanturri.

Defining data
Ambient
ℙ3P3 × Gr(2, 7)
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
2 copies of E^(w5) + E^(w1 + w5) + E^(w3 + w4)
Bundle
2 copies of O_2(1) + O_1(1) box O_2(1) + Q_1 box U_2^v
ZL62 label
30601.44wFw3
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
$\mathrm{h}^0(X, \omega_X^\vee)$ = 33
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 129
$\chi(X, \mathrm{T}_X)$ = -23

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 129
$\mathrm{c}_1^2\mathrm{c}_2$ = 126
$\mathrm{c}_2^2$ = 118
$\mathrm{c}_1\mathrm{c}_3$ = 22
$\mathrm{c}_4$ = 31
Hodge diamond
1
00
0 2 0
0 0 0 0
0 1 23 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 + 23 0 x0 + 23 x2
$\mathrm{HH}^{ 3 }$ = x1 + 19 0 0 x1 19
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 26 0 0 x1 - x2 - 7 0 33
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 28t + 71t^{2} + 28t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(361)
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) ^{3} \left( 25439t^{7} + 44164t^{6} + 33728t^{5} + 12856t^{4} + 1931t^{3} + 55t^{2} + 3t - 1 \right) \left( 1337767014416036t^{14} + 5922380541042514t^{13} + 11994844693000854t^{12} + 14767833824336317t^{11} + 12387188117642713t^{10} + 7480333592667915t^{9} + 3323491290119128t^{8} + 1083694856390618t^{7} + 253935038025638t^{6} + 41376358573906t^{5} + 4554074828463t^{4} + 339464571979t^{3} + 15980632255t^{2} + 97281082t + 844950 \right)D^{6} + \left( 2t + 1 \right) ^{2} \left( 1429321113348640671768t^{22} + 9046151601073070750808t^{21} + 27056658995808829330370t^{20} + 50871692905931876774196t^{19} + 67460944201400102170035t^{18} + 67050145306872877004664t^{17} + 51748432384109718489967t^{16} + 31658785228104531933050t^{15} + 15515012066832946466027t^{14} + 6108108037746152831705t^{13} + 1925513906278798070736t^{12} + 482169159091501033642t^{11} + 94855545387020926460t^{10} + 14493325006951053027t^{9} + 1706584133527915420t^{8} + 154735672877578756t^{7} + 10850166321831535t^{6} + 601046317152011t^{5} + 30023570929137t^{4} + 1587815851815t^{3} + 63381599482t^{2} + 285083646t + 1689900 \right)D^{5} + \left( 2t + 1 \right) \left( 23822018555810677862800t^{23} + 156312024523153006882520t^{22} + 484803729524152264596548t^{21} + 946044626858601820901662t^{20} + 1304281925287282997601146t^{19} + 1351299602145261941306715t^{18} + 1091117706248452616222401t^{17} + 701721689131517170876255t^{16} + 363730687946715147755124t^{15} + 152661816581327738842717t^{14} + 51837706010319202277486t^{13} + 14169365345529640227019t^{12} + 3093091978596059636694t^{11} + 534020619746886033653t^{10} + 72117444400578815016t^{9} + 7494239936652264530t^{8} + 575710258964782119t^{7} + 28950593337459468t^{6} + 585842300114941t^{5} - 20605615359849t^{4} - 1499610246084t^{3} - 57976923889t^{2} - 241701204t - 844950 \right)D^{4} + t \left( 200104955868809694047520t^{23} + 1371043508429486930725440t^{22} + 4445011311881511356431360t^{21} + 9080558662284183919750096t^{20} + 13131400954062098220543882t^{19} + 14305888864853275589729200t^{18} + 12185319507926998285839249t^{17} + 8299964299327646750630328t^{16} + 4579842988963622927852565t^{15} + 2059620607309692141831786t^{14} + 755626102572370686276977t^{13} + 225534374447131938556975t^{12} + 54473693718939960452508t^{11} + 10574269336715500268348t^{10} + 1637188476000195661432t^{9} + 200331532502447820213t^{8} + 19051085373468382740t^{7} + 1350910645209856434t^{6} + 64779238911002935t^{5} + 1721592880893231t^{4} + 19726747118317t^{3} + 197533841821t^{2} + 1652996654t + 1342750 \right)D^{3} + 2 t ^{2} \left( 221068332197923090566784t^{22} + 1479151301449985259635776t^{21} + 4677720913939472299198528t^{20} + 9313911344329469259058208t^{19} + 13121682550206308034532396t^{18} + 13921946936291643954959664t^{17} + 11541844193756308500324713t^{16} + 7643044076176253229483098t^{15} + 4092166544320756853782149t^{14} + 1780844321500935920402836t^{13} + 630180894409988703078317t^{12} + 180806384233218565928935t^{11} + 41853218152269383645189t^{10} + 7770314816512193177868t^{9} + 1149464628315510382532t^{8} + 134217971307679104749t^{7} + 12111425958856676813t^{6} + 801128445435784553t^{5} + 34621454944557758t^{4} + 799821098178328t^{3} + 10031981379267t^{2} + 138312619353t + 691667040 \right)D^{2} + 8 t ^{2} \left( 60031486760642908214256t^{22} + 394596585000674362241472t^{21} + 1224845330522656796161920t^{20} + 2392379466716412504390168t^{19} + 3305188449305286168244121t^{18} + 3437874307667781530790208t^{17} + 2792542298413190632588015t^{16} + 1809769809311310323498976t^{15} + 946495459024659294762010t^{14} + 401302739933954443577801t^{13} + 137932650017986208575608t^{12} + 38319744113191746138086t^{11} + 8566184430096931060800t^{10} + 1533071184144925657242t^{9} + 218379424577499256629t^{8} + 24503276520976071973t^{7} + 2109170783503534820t^{6} + 130531178475072180t^{5} + 5079485526938964t^{4} + 101229642467402t^{3} + 1357118119724t^{2} + 18294047054t + 69285900 \right)D + 48 t ^{2} \left( 4083774609567544776480t^{22} + 26518398055863723048960t^{21} + 81263283912851788668288t^{20} + 156625523134520719489008t^{19} + 213466005945421559631646t^{18} + 218980452994665065745776t^{17} + 175335011600789799593636t^{16} + 111892049892336662499456t^{15} + 57528549619842863454767t^{14} + 23925075469043324233456t^{13} + 8044959400652526660342t^{12} + 2180677608974470579642t^{11} + 474530257325475292383t^{10} + 82536888554199438255t^{9} + 11413570592015885853t^{8} + 1240423909755201521t^{7} + 102670894821036274t^{6} + 6001685101203438t^{5} + 212898616194882t^{4} + 3724169312806t^{3} + 52709176786t^{2} + 595680580t + 1689900 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 7)

|λ| = 16.60439 λ ≈ 16.60439; multiplicity 1; Jordan blocks 1 λ ≈ -3.211998 - 9.763137i; multiplicity 1; Jordan blocks 1 λ ≈ -3.211998 + 9.763137i; multiplicity 1; Jordan blocks 1 λ ≈ -3.267669; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 3; Jordan blocks 1, 2 λ ≈ -1.727228; multiplicity 1; Jordan blocks 1 λ ≈ -1.092749 - 1.172849i; multiplicity 1; Jordan blocks 1 λ ≈ -1.092749 + 1.172849i; multiplicity 1; Jordan blocks 1 3
Sector dimension
10
Spectral radius
$16.6043912$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$16.60439$11
$-3.211998 - 9.763137\mathrm{i}$11
$-3.211998 + 9.763137\mathrm{i}$11
$-3.267669$11
$-2$31, 2
$-1.727228$11
$-1.092749 - 1.172849\mathrm{i}$11
$-1.092749 + 1.172849\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{3} \left(\lambda^{7} - 3 \lambda^{6} - 55 \lambda^{5} - 1931 \lambda^{4} - 12856 \lambda^{3} - 33728 \lambda^{2} - 44164 \lambda - 25439\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda^{7} - 3 \lambda^{6} - 55 \lambda^{5} - 1931 \lambda^{4} - 12856 \lambda^{3} - 33728 \lambda^{2} - 44164 \lambda - 25439\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 8 & 84 & 42 & 42 & 96 & 232 & 384 & 500 \\ 1 & 0 & 5 & -6 & -6 & 2 & -2 & -3 & 3 & 120 \\ 1 & 0 & 0 & 14 & 10 & 4 & 14 & 69 & 120 & 160 \\ 0 & 1 & 1 & 0 & -1 & 3 & 4 & -10 & -18 & 0 \\ 0 & 1 & 0 & 5 & 1 & 2 & 5 & 6 & 8 & 0 \\ 0 & 0 & 1 & 0 & 1 & -2 & 0 & 4 & 10 & 30 \\ 0 & 0 & 1 & 0 & 0 & 0 & -2 & 10 & 18 & 30 \\ 0 & 0 & 0 & -\frac{3}{7} & -\frac{18}{7} & \frac{24}{7} & \frac{17}{7} & -\frac{60}{7} & -\frac{80}{7} & \frac{20}{7} \\ 0 & 0 & 0 & \frac{18}{7} & \frac{17}{7} & -\frac{4}{7} & \frac{10}{7} & \frac{45}{7} & \frac{60}{7} & \frac{20}{7} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{5} & \frac{13}{5} & 0\end{pmatrix}\notag\end{equation}$$