Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-80-24-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ3P3
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_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + 3w3)
Bundle
O_1(1) box O_2(3)
ZL62 label
2030.x312
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(1;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 80
$\chi(X, \mathrm{T}_X)$ = -36

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 80
$\mathrm{c}_1^2\mathrm{c}_2$ = 116
$\mathrm{c}_2^2$ = 123
$\mathrm{c}_1\mathrm{c}_3$ = 2
$\mathrm{c}_4$ = 35
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 29 0 0
0 0 0 0
0 2 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 36 0 36 0
$\mathrm{HH}^{ 3 }$ = x0 + 12 0 0 x0 12
$\mathrm{HH}^{ 4 }$ = x0 + 10 0 0 x0 - 18 4 24
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 40t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(696)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(694)

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( 6t + 1 \right) \left( 1139t^{3} + 360t^{2} + 23t - 1 \right) \left( 74493t^{5} + 29876t^{4} + 2924t^{3} - 93t^{2} - 10t + 1 \right) \left( 494747309064667044240t^{13} + 485194468232890731240t^{12} + 272824830637220407266t^{11} + 117781673922493716531t^{10} + 34097132845888543901t^{9} + 5988393824642622930t^{8} + 617668147486231472t^{7} + 36784994626322914t^{6} + 1320804923625276t^{5} + 37738128376491t^{4} + 976808359550t^{3} + 17425846190t^{2} - 611507197t + 474320 \right)D^{6} + \left( 5289238793669251904552612904480t^{22} + 9218302733989897847176041170640t^{21} + 8124953687316301622890549018716t^{20} + 4940561483449614953445466366362t^{19} + 2256556713741896341968750741930t^{18} + 774610145634265957581064352913t^{17} + 196387115407189538171421058030t^{16} + 36231706046018399132893363787t^{15} + 4792476197979705897624036236t^{14} + 444019428230834129781626600t^{13} + 27508620420921886639207682t^{12} + 1008050682218231392089933t^{11} + 9770479996750303697635t^{10} - 1066918827502741631606t^{9} - 55635042267318609630t^{8} + 277134400049510325t^{7} + 150284691346320362t^{6} + 8010494538666306t^{5} + 263095579634594t^{4} + 3567477712865t^{3} + 54390822761t^{2} - 1808908311t + 948640 \right)D^{5} + \left( 44076989947243765871271774204000t^{22} + 72529913384242927916139672778800t^{21} + 61536990436137414094044738435084t^{20} + 36742948436196494244095081814450t^{19} + 16499360782263560549696335409733t^{18} + 5498284616940581090232308997927t^{17} + 1335510257071102532140051034135t^{16} + 234457183481035875693834930945t^{15} + 29605075181884838194138249630t^{14} + 2671607599183101200336547120t^{13} + 170787751448882639143555058t^{12} + 7658562553444032211071496t^{11} + 233038852407758472469571t^{10} + 3176642577792368266152t^{9} - 204851447099772048002t^{8} - 21044846356315182267t^{7} - 940459238141487137t^{6} - 20080918919036233t^{5} - 274833413863001t^{4} - 5864468979215t^{3} - 61160574011t^{2} + 1131784871t - 474320 \right)D^{4} + t \left( 185123357778423816659341451656800t^{21} + 290471025197099768573706413648400t^{20} + 239216228561512293823114380346500t^{19} + 140942856014978468926619268363042t^{18} + 62269501240340326228237304529186t^{17} + 20137997923170303209612871578727t^{16} + 4688992861354428954825272829574t^{15} + 783846345841931264743943458725t^{14} + 94277294042422303565169559956t^{13} + 8191153716198005020683296292t^{12} + 519313685087710578578013024t^{11} + 24650708160440703286643255t^{10} + 911553400169109325839739t^{9} + 26188589845824700438530t^{8} + 463577282739766354462t^{7} - 3895645517582536921t^{6} - 461551135379133848t^{5} - 3076910264210308t^{4} + 68265816916548t^{3} - 3158331127001t^{2} + 25372952003t + 3480323 \right)D^{3} + 2 t ^{2} \left( 204517233355211073642701032306560t^{20} + 308937306620282154616431668249640t^{19} + 248712846195502742224520963091924t^{18} + 145130643377296791358770966409137t^{17} + 63184104352959392351717643793878t^{16} + 19878943676597969375460186218073t^{15} + 4456111289881282272090668479462t^{14} + 713095469474264000467589666990t^{13} + 82092556334766221439887772458t^{12} + 6879298389259051917275662024t^{11} + 428927658278114724380815405t^{10} + 20668653058604271948691546t^{9} + 797693345665864524616828t^{8} + 23812330911995162413084t^{7} + 420095489069464969278t^{6} - 3092180173320362515t^{5} - 314316791411722055t^{4} - 1157964193130875t^{3} + 59112963694910t^{2} - 711098585492t + 6194182400 \right)D^{2} + 4 t ^{2} \left( 111074014667054289995604870994080t^{20} + 163023265445868531517114588625640t^{19} + 129074766493827963548272163967576t^{18} + 74805777105871668971500163401065t^{17} + 32169671442891740776510961726058t^{16} + 9889884997154489370230203094652t^{15} + 2148137100532990451277538548310t^{14} + 331587886910308435352072179024t^{13} + 36819800278149376573987410022t^{12} + 2995037859358903287116843818t^{11} + 184008609310525647307081535t^{10} + 8915871063463513258289204t^{9} + 349445179273351166780475t^{8} + 10428915450409902300636t^{7} + 172808255696970130099t^{6} - 1755711800597480463t^{5} - 122941582116303112t^{4} - 349571099200396t^{3} + 17516232976220t^{2} - 34413978956t + 732350080 \right)D + 16 t ^{2} \left( 11334083129291254081184170509600t^{20} + 16306766162919782235649906723800t^{19} + 12765210638405972343791405273856t^{18} + 7364476087651100353734022400127t^{17} + 3138212859357314162408200625502t^{16} + 948230183041091498075141644992t^{15} + 201158229581225742503168409810t^{14} + 30220604636280976533123419754t^{13} + 3265489054120270268918813190t^{12} + 259720420968081713708500042t^{11} + 15772829493771013603821237t^{10} + 765042252799498756260158t^{9} + 30100687437526903405241t^{8} + 890874274948582946850t^{7} + 14146970324956832843t^{6} - 156925224880876809t^{5} - 9023576221633970t^{4} - 11012213502318t^{3} + 944768057296t^{2} + 5037017524t + 26561920 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ3P3

|λ| = 34.42032 λ ≈ 34.42032; multiplicity 1; Jordan blocks 1 λ ≈ 14.25293 - 10.8438i; multiplicity 1; Jordan blocks 1 λ ≈ 14.25293 + 10.8438i; multiplicity 1; Jordan blocks 1 λ ≈ -6.56906 - 0.3416124i; multiplicity 1; Jordan blocks 1 λ ≈ -6.56906 + 0.3416124i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -5.710158 - 0.6964285i; multiplicity 1; Jordan blocks 1 λ ≈ -5.710158 + 0.6964285i; multiplicity 1; Jordan blocks 1 λ ≈ -5.367735; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$34.4203168$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$34.42032$11
$14.25293 - 10.8438\mathrm{i}$11
$14.25293 + 10.8438\mathrm{i}$11
$-6.56906 - 0.3416124\mathrm{i}$11
$-6.56906 + 0.3416124\mathrm{i}$11
$-6$11
$-5.710158 - 0.6964285\mathrm{i}$11
$-5.710158 + 0.6964285\mathrm{i}$11
$-5.367735$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda^{3} - 23 \lambda^{2} - 360 \lambda - 1139\right) \left(\lambda^{5} - 10 \lambda^{4} - 93 \lambda^{3} + 2924 \lambda^{2} + 29876 \lambda + 74493\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda^{3} - 23 \lambda^{2} - 360 \lambda - 1139\right) \left(\lambda^{5} - 10 \lambda^{4} - 93 \lambda^{3} + 2924 \lambda^{2} + 29876 \lambda + 74493\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 108 & 54 & 54 & 777 & 1152 & 1296 & 3785 \\ 2 & 0 & 11 & 108 & 0 & 156 & 780 & 45 & 240 \\ 1 & 0 & 9 & 2 & 6 & 108 & 90 & 135 & 432 \\ 0 & 1 & 2 & 9 & 0 & 14 & 108 & 0 & 12 \\ 0 & 2 & 0 & 11 & 0 & 6 & 156 & 108 & 255 \\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 6 & 18 \\ 0 & 0 & 0 & 1 & 0 & \frac{7}{3} & 0 & 0 & \frac{2}{3} \\ 0 & 0 & 0 & 2 & 1 & -\frac{1}{9} & 14 & 9 & \frac{322}{9} \\ 0 & 0 & 0 & 0 & 0 & 0 & 7 & 3 & 0\end{pmatrix}\notag\end{equation}$$