Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-336-72-a

·

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

Defining data
Ambient
Fl(1, 2, 3) × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4ω1ω2ω3ω4
Bundle
E^(w1 + w3)
Bundle
O_1(1,0) box O_2(1)
ZL62 label
2022.T
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2,2;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1,0;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 72
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 336
$\chi(X, \mathrm{T}_X)$ = 8

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 336
$\mathrm{c}_1^2\mathrm{c}_2$ = 180
$\mathrm{c}_2^2$ = 96
$\mathrm{c}_1\mathrm{c}_3$ = 60
$\mathrm{c}_4$ = 12
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 4 0 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = x0 0 x0
$\mathrm{HH}^{ 2 }$ = x0 + x1 - 8 0 x0 - 8 x1
$\mathrm{HH}^{ 3 }$ = x1 + 53 0 0 x1 - 37 90
$\mathrm{HH}^{ 4 }$ = 72 0 0 0 0 72
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 67t + 200t^{2} + 67t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(165)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(230)

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( 3t + 1 \right) \left( 9t^{2} - 3t + 1 \right) \left( 25t^{2} + 5t - 1 \right) \left( 139t^{4} - 65t^{3} + 20t^{2} + 8t + 1 \right) \left( 29283936811410t^{13} + 48366656484408t^{12} - 14151690702149t^{11} + 8343304521015t^{10} + 18165893913504t^{9} + 10062820281450t^{8} + 577942986748t^{7} - 277799626928t^{6} - 53175766677t^{5} + 5913337423t^{4} - 7861288t^{3} + 31829332t^{2} + 463520t + 11648 \right)D^{6} + \left( 57698872797941408250t^{22} + 87335599875604186500t^{21} - 52029724315243739355t^{20} + 29710450598214525174t^{19} + 44703040649299600230t^{18} + 13104771285739949900t^{17} - 2522325971877423519t^{16} + 2239452959549886477t^{15} + 1949107233425402551t^{14} + 189851030321939295t^{13} - 60189770138070323t^{12} - 8588604179005175t^{11} + 1462702469613063t^{10} + 22944583245019t^{9} + 25223882093840t^{8} + 11746310764311t^{7} - 2198904547255t^{6} - 251956507313t^{5} + 42588410034t^{4} + 104355800t^{3} + 129071728t^{2} + 1320672t + 23296 \right)D^{5} + \left( 480823939982845068750t^{22} + 777658533349952227500t^{21} - 422010463568730203925t^{20} + 243793403032505967843t^{19} + 401698161976020394068t^{18} + 162547555139549490774t^{17} - 23577691125435765987t^{16} + 5607759481000442972t^{15} + 10054840776970812852t^{14} + 1200861254080338006t^{13} - 459171617486184888t^{12} - 111250170796728460t^{11} + 2905847530326760t^{10} - 779885690540084t^{9} - 261099983905679t^{8} + 15599611742755t^{7} + 2938784948116t^{6} + 1104185799754t^{5} - 22790833599t^{4} + 493026428t^{3} - 126056916t^{2} - 1662944t - 11648 \right)D^{4} + t \left( 2019460547927949288750t^{21} + 3430710003545526577500t^{20} - 1718004353451632480925t^{19} + 1054325500525628574870t^{18} + 1860415480985858942202t^{17} + 892301178772224362176t^{16} - 96250519239799647561t^{15} - 13936070238532562197t^{14} + 23116221761622045281t^{13} + 4231165715539396491t^{12} - 1347964927984343189t^{11} - 329030461450582985t^{10} + 21456078572507373t^{9} + 2062344948153063t^{8} + 278778167805590t^{7} + 160197074225t^{6} - 3857920788347t^{5} + 488537209479t^{4} + 18879219580t^{3} + 689329816t^{2} + 11563736t + 2080 \right)D^{3} + 2 t ^{2} \left( 2231023081520401119000t^{20} + 3929196713521446418500t^{19} - 1841892186682013559795t^{18} + 1215412462004087646432t^{17} + 2249092744515091488600t^{16} + 1188975021020477506366t^{15} - 94054230985065907191t^{14} - 42602478800213283577t^{13} + 11451757288999607108t^{12} + 4218146238973580275t^{11} - 1028185588596912435t^{10} - 238756455026837249t^{9} + 23711096835578590t^{8} + 2181938296708792t^{7} + 248319878398870t^{6} - 31987718056380t^{5} - 2370960684868t^{4} + 245248914652t^{3} + 8193849058t^{2} + 370221124t + 4711392 \right)D^{2} + 8 t ^{2} \left( 605838164378384786625t^{20} + 1094656702446049482000t^{19} - 487615983671872795140t^{18} + 343416441459119438892t^{17} + 655722865669831594029t^{16} + 367441065449722981803t^{15} - 21425613863146877034t^{14} - 16319326865592031231t^{13} + 667091733182311827t^{12} + 1100343977515214217t^{11} - 197263802252585254t^{10} - 44831463397140670t^{9} + 5608029125395754t^{8} + 444934169689492t^{7} + 48864925063284t^{6} - 9538848722054t^{5} - 310879423028t^{4} + 32350894520t^{3} + 1146768400t^{2} + 54550016t + 512512 \right)D + 16 t ^{2} \left( 123640441709874446250t^{20} + 227215286294085405000t^{19} - 97665831566951833350t^{18} + 72198351811756451052t^{17} + 140555222709883135974t^{16} + 81507308973976773684t^{15} - 3651739615849484511t^{14} - 3931140873021819282t^{13} - 169688283455350788t^{12} + 222383213767986870t^{11} - 29932812222652153t^{10} - 6935706890230137t^{9} + 967169976008948t^{8} + 66513683722997t^{7} + 7800707812627t^{6} - 1747604448881t^{5} - 28521869952t^{4} + 3539939060t^{3} + 153206560t^{2} + 6888128t + 46592 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 2, 3) × ℙ2P2

|λ| = 8.09017 λ ≈ 8.09017; multiplicity 1; Jordan blocks 1 λ ≈ -5.199066 - 3.81628i; multiplicity 1; Jordan blocks 1 λ ≈ -5.199066 + 3.81628i; multiplicity 1; Jordan blocks 1 λ ≈ -3.09017; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.199066 - 1.379872i; multiplicity 1; Jordan blocks 1 λ ≈ 1.199066 + 1.379872i; multiplicity 1; Jordan blocks 1 2 2 2
Sector dimension
12
Spectral radius
$8.09016994$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.09017$11
$-5.199066 - 3.81628\mathrm{i}$11
$-5.199066 + 3.81628\mathrm{i}$11
$-3.09017$11
$-3$21, 1
$1.5 - 2.598076\mathrm{i}$21, 1
$1.5 + 2.598076\mathrm{i}$21, 1
$1.199066 - 1.379872\mathrm{i}$11
$1.199066 + 1.379872\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda^{2} - 3 \lambda + 9\right)^{2} \left(\lambda^{2} - 5 \lambda - 25\right) \left(\lambda^{4} + 8 \lambda^{3} + 20 \lambda^{2} - 65 \lambda + 139\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{2} - 3 \lambda + 9\right) \left(\lambda^{2} - 5 \lambda - 25\right) \left(\lambda^{4} + 8 \lambda^{3} + 20 \lambda^{2} - 65 \lambda + 139\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 2 & 2 & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 5 \\ 1 & -1 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 3 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 0 & 0 \\ 0 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 \\ 0 & 2 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 \\ 0 & 3 & 1 & -2 & 0 & -1 & -1 & 0 & 2 & 0 & 2 & -3 \\ 0 & 0 & 2 & 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 2 & -2 & -1 & 0 & -1 & 2 \\ 0 & 0 & 0 & 0 & 2 & 2 & 0 & 5 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 & 2 & 3 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 2 & 0\end{pmatrix}\notag\end{equation}$$