Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-325-70-a

·

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

Defining data
Ambient
Fl(2, 3, 5)
node 1: fundamental weight ω_1node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1) + 2 copies of E^(w3)
Bundle
U_3^v + 2 copies of O(0,1)
ZL62 label
45.220
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 70
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 325
$\chi(X, \mathrm{T}_X)$ = 4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 325
$\mathrm{c}_1^2\mathrm{c}_2$ = 178
$\mathrm{c}_2^2$ = 97
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 10
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 4 0 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 + x1 - 4 0 x0 - 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 57 0 0 x1 - 33 90
$\mathrm{HH}^{ 4 }$ = 70 0 0 0 0 70
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 65t + 193t^{2} + 65t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(25)
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( 7t^{3} - 3t^{2} + 8t - 1 \right) \left( 1477t^{7} + 1005t^{6} + 1616t^{5} + 1389t^{4} + 506t^{3} + 107t^{2} + 12t + 1 \right) \left( 7496856720920t^{17} + 200613321910040t^{16} - 70546594246464t^{15} - 342304078653520t^{14} - 840326083288565t^{13} - 614840058096784t^{12} - 472782445912869t^{11} + 13275901788905t^{10} + 130148952263031t^{9} + 105114474223080t^{8} + 43075117088743t^{7} + 1018648635829t^{6} - 2171553495671t^{5} - 509730382420t^{4} - 30361695603t^{3} - 2792157987t^{2} - 18222284t - 488800 \right)D^{6} + \left( 1627710034389429480t^{27} + 45962975828335564880t^{26} - 5412425217496773448t^{25} - 31167486323070061664t^{24} - 232944386349238970455t^{23} - 316597628319773093988t^{22} - 503642233908832874698t^{21} - 405791479708831413318t^{20} - 336885130746626832632t^{19} - 140689362177404921905t^{18} - 14220342641955059574t^{17} + 45894809768384965232t^{16} + 57803136183118085878t^{15} + 36544452098342343484t^{14} + 17069398508613114025t^{13} + 5071987603327407636t^{12} + 342125207250728236t^{11} - 291621938128156098t^{10} - 111042634508453117t^{9} - 18529285519553730t^{8} - 1750604179837157t^{7} - 155835945912525t^{6} - 25824318495171t^{5} - 3092686638536t^{4} - 152101966135t^{3} - 11054683612t^{2} - 50756452t - 977600 \right)D^{5} + \left( 13564250286578579000t^{27} + 396331824406948379600t^{26} - 78893589479053701352t^{25} - 526893279709113594808t^{24} - 2281841748165579209217t^{23} - 2733479284671613727788t^{22} - 3753006948143200955614t^{21} - 2400950784190525097213t^{20} - 1612167091213915557524t^{19} - 197783118694299851522t^{18} + 440641832511337899780t^{17} + 554971801542295941623t^{16} + 412421999233897733793t^{15} + 186775420547959073699t^{14} + 67835220824780691106t^{13} + 14367878691731740007t^{12} - 1605421393150827083t^{11} - 1923982249733535818t^{10} - 546649678021859352t^{9} - 83852472631564444t^{8} - 9213762011373906t^{7} - 639522694101200t^{6} - 6784170138296t^{5} + 3991073119544t^{4} + 135379044521t^{3} + 10647900573t^{2} + 46327352t + 488800 \right)D^{4} + t \left( 56969851203630031800t^{26} + 1708506847772884812800t^{25} - 456798209604650851000t^{24} - 3098215554330737570720t^{23} - 10717889782117262722389t^{22} - 11753557372154806343244t^{21} - 14353730920166049400170t^{20} - 7177003353117620922942t^{19} - 3547641615878299105084t^{18} + 1252785177709389228377t^{17} + 2683094311142088952954t^{16} + 2293830886640946975448t^{15} + 1379714681254536145680t^{14} + 480976057487252674884t^{13} + 130223886769553663421t^{12} + 12992547030492833912t^{11} - 9938862048515896116t^{10} - 5490364314098316334t^{9} - 1310635149058121853t^{8} - 171069495005323102t^{7} - 16869164318266595t^{6} - 1226677454536351t^{5} - 50412257804251t^{4} - 651690674136t^{3} - 13242372391t^{2} - 21235640t - 106784 \right)D^{3} + 2 t ^{2} \left( 62938121329724606560t^{25} + 1924610376512458637380t^{24} - 619325814914032323272t^{23} - 4170901439481609834512t^{22} - 12830754506601609058228t^{21} - 13273046570477406044264t^{20} - 14862885393753387023475t^{19} - 5705635799601200680076t^{18} - 1496230678291244298723t^{17} + 2698588371088569832699t^{16} + 3266603893551855113209t^{15} + 2285940892370315468915t^{14} + 1224750362802580997699t^{13} + 339325948025499403496t^{12} + 59516457819568655513t^{11} - 6598779068640589848t^{10} - 10022235218257027628t^{9} - 4081009163557875220t^{8} - 877741255511250236t^{7} - 106167623183612224t^{6} - 10581611157767129t^{5} - 729480861331989t^{4} - 26289692522512t^{3} - 429475475685t^{2} - 9073918975t - 7391150 \right)D^{2} + 4 t ^{2} \left( 34181910722178019080t^{25} + 1060035051487129902380t^{24} - 384381668601574066528t^{23} - 2562509800504396790024t^{22} - 7377162715490204545979t^{21} - 7356688503639450331732t^{20} - 7760486741753880715567t^{19} - 2283814253580736160328t^{18} + 91887437114807754537t^{17} + 1908624017212180164142t^{16} + 1820527288787703444783t^{15} + 1108238642518251299363t^{14} + 555752798919856286517t^{13} + 125658128565381868670t^{12} + 9093804561910600996t^{11} - 8303015950523724982t^{10} - 4618625521347683792t^{9} - 1558076344752135200t^{8} - 310549500217341335t^{7} - 36043714117513570t^{6} - 3685963817612378t^{5} - 232588296306776t^{4} - 7313639818213t^{3} - 146379460513t^{2} - 2842234384t - 977600 \right)D + 48 t ^{3} \left( 1162650024563878200t^{24} + 36395078940994690700t^{23} - 14211164376101936656t^{22} - 93981749658108239804t^{21} - 260573233101578107765t^{20} - 254238163113020889012t^{19} - 258212179512679648451t^{18} - 59973331295814623852t^{17} + 23238011303224573620t^{16} + 74293059622595390070t^{15} + 62218355762941507184t^{14} + 34329617649027296585t^{13} + 16591913860829181378t^{12} + 3170564773209342390t^{11} - 98248812006525109t^{10} - 351365395895989618t^{9} - 135342737460882032t^{8} - 40058472263315863t^{7} - 7528415162983102t^{6} - 858707026879448t^{5} - 90327159070870t^{4} - 5154575912047t^{3} - 143430988046t^{2} - 3423145329t - 59389200 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(2, 3, 5)

|λ| = 7.729032 λ ≈ 7.729032; multiplicity 1; Jordan blocks 1 λ ≈ -2.220777 - 6.529031i; multiplicity 1; Jordan blocks 1 λ ≈ -2.220777 + 6.529031i; multiplicity 1; Jordan blocks 1 λ ≈ -2.880175 - 2.606355i; multiplicity 1; Jordan blocks 1 λ ≈ -2.880175 + 2.606355i; multiplicity 1; Jordan blocks 1 λ ≈ -2.073176; multiplicity 1; Jordan blocks 1 λ ≈ 0.1375395 - 0.9868481i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1375395 + 0.9868481i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1354842 - 0.9419768i; multiplicity 1; Jordan blocks 1 λ ≈ 0.1354842 + 0.9419768i; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$7.72903154$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$7.729032$11
$-2.220777 - 6.529031\mathrm{i}$11
$-2.220777 + 6.529031\mathrm{i}$11
$-2.880175 - 2.606355\mathrm{i}$11
$-2.880175 + 2.606355\mathrm{i}$11
$-2.073176$11
$0.1375395 - 0.9868481\mathrm{i}$11
$0.1375395 + 0.9868481\mathrm{i}$11
$0.1354842 - 0.9419768\mathrm{i}$11
$0.1354842 + 0.9419768\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda^{3} - 8 \lambda^{2} + 3 \lambda - 7\right) \left(\lambda^{7} + 12 \lambda^{6} + 107 \lambda^{5} + 506 \lambda^{4} + 1389 \lambda^{3} + 1616 \lambda^{2} + 1005 \lambda + 1477\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda^{3} - 8 \lambda^{2} + 3 \lambda - 7\right) \left(\lambda^{7} + 12 \lambda^{6} + 107 \lambda^{5} + 506 \lambda^{4} + 1389 \lambda^{3} + 1616 \lambda^{2} + 1005 \lambda + 1477\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 6 & 6 & 3 & 3 & 4 & 4 & 0 \\ 2 & 0 & 2 & -2 & 0 & 0 & 0 & 0 & 3 & 4 \\ 1 & 0 & -1 & 2 & 0 & 0 & 0 & 3 & 3 & 0 \\ 0 & 2 & 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 3 & 0 & -1 & 0 & 0 & 0 & 0 & 3 \\ 0 & 3 & 2 & 0 & 1 & 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & 9 & 2 & 6 & -1 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 2 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$