Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-486-100-a

·

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

Defining data
Ambient
Fl(1, 3, 6)
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
2 copies of E^(w5) + E^(2w3)
Bundle
2 copies of wedge^2 Q_1^v ot wedge^2 U_2^v ot U_3^v + O(0,2)
ZL62 label
54.447
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3,5) \\ \det\mathcal{E} &\cong \mathcal{O}(0,4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 100
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 486
$\chi(X, \mathrm{T}_X)$ = 18

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 486
$\mathrm{c}_1^2\mathrm{c}_2$ = 216
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 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 + x2 - 18 0 x0 - 18 x2
$\mathrm{HH}^{ 3 }$ = x1 + 156 0 0 x1 156
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 185 0 0 x1 - x2 + 85 0 100
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 95t + 294t^{2} + 95t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(9)
Coates et al. (2020) Obro(4,74)
Coates–Kasprzyk–Prince (2015) CKP(13)

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( 5t + 1 \right) \left( 9t^{2} + 3t + 1 \right) \left( 31t^{2} + t - 1 \right) \left( 101t^{3} + 36t^{2} + 4t + 1 \right) \left( 2321065005451776t^{13} + 4827971533187232t^{12} + 4376915001078192t^{11} + 2214147013707456t^{10} + 683275522879704t^{9} + 135549509831424t^{8} + 19112801125905t^{7} + 2459944478520t^{6} + 324421754395t^{5} + 30797468505t^{4} + 1386190584t^{3} + 18127179t^{2} + 605556t + 26244 \right)D^{6} + \left( 20602666598417062709760t^{22} + 54714084451537191009984t^{21} + 65407480973309057045904t^{20} + 45617285388402057761280t^{19} + 20122122829637922610680t^{18} + 5539323773319284184624t^{17} + 745430926057842424383t^{16} - 81065328855778216056t^{15} - 68396736063038799285t^{14} - 19405503530993036565t^{13} - 4076802255314078712t^{12} - 810433062845586119t^{11} - 154945594410415737t^{10} - 24681924544924536t^{9} - 2988520801304715t^{8} - 289809904695155t^{7} - 27749531952744t^{6} - 2727081888805t^{5} - 195790541418t^{4} - 6903766080t^{3} - 65144844t^{2} - 1396764t - 52488 \right)D^{5} + \left( 171688888320142189248000t^{22} + 454153971481202857452480t^{21} + 540577900181831450468400t^{20} + 376199257511757194956800t^{19} + 168031565441920532710440t^{18} + 49633140263109584558688t^{17} + 9439947329849391581721t^{16} + 964886799386423758896t^{15} - 24059401451153395803t^{14} - 30397738104479158758t^{13} - 7632467768335833576t^{12} - 1443903899437923104t^{11} - 204016037778065898t^{10} - 16401926524282578t^{9} + 35886695825214t^{8} + 161218521912382t^{7} + 21759299578016t^{6} + 2383095187220t^{5} + 199753663113t^{4} + 8024222934t^{3} + 92011761t^{2} + 1213056t + 26244 \right)D^{4} + t \left( 721093330944597194841600t^{21} + 1901517463691748678119040t^{20} + 2255450589198996267389040t^{19} + 1563805259038342222519680t^{18} + 699462119396162221266888t^{17} + 211505550495486451231824t^{16} + 44429639808940512562953t^{15} + 6716797515589939000824t^{14} + 786055610949373129617t^{13} + 70103770975266270435t^{12} - 866017403377733736t^{11} - 2540354001986858605t^{10} - 635464265144895591t^{9} - 86594409271279776t^{8} - 7056283687270065t^{7} - 340695455923881t^{6} - 10800088250370t^{5} - 392930856779t^{4} - 14144570856t^{3} - 230363514t^{2} - 9936t - 1944 \right)D^{3} + 2 t ^{2} \left( 796636441805459758110720t^{20} + 2095712455440639778375824t^{19} + 2478966783345321935981616t^{18} + 1712236097495292288193440t^{17} + 763811588906769030022512t^{16} + 232356834724855302303768t^{15} + 50552007040516540268133t^{14} + 8524420463530536095400t^{13} + 1263044762183835231858t^{12} + 168835558016013730560t^{11} + 15233986188719913132t^{10} - 193231848415986276t^{9} - 305617954439923692t^{8} - 47907401045088651t^{7} - 3957900593277582t^{6} - 194395843165203t^{5} - 7031207621705t^{4} - 305112708240t^{3} - 10509144552t^{2} - 142742466t + 2916 \right)D^{2} + 36 t ^{3} \left( 48072888729639812989440t^{19} + 126243783997314517148976t^{18} + 149024766148613539374816t^{17} + 102598586765635249274400t^{16} + 45604315594650729361320t^{15} + 13869149702634768491496t^{14} + 3055595871122497470114t^{13} + 537543854223920678816t^{12} + 85892827356634929765t^{11} + 12516428982364071147t^{10} + 1345993016595950628t^{9} + 62215731874356027t^{8} - 6564398782941899t^{7} - 1449125314393779t^{6} - 126349095513543t^{5} - 6611219455950t^{4} - 281168510823t^{3} - 13529041929t^{2} - 446088222t - 5196312 \right)D + 432 t ^{3} \left( 1635132269715639897600t^{19} + 4288913479836074654640t^{18} + 5055768563338791840960t^{17} + 3472479322438069031040t^{16} + 1538790168061781917848t^{15} + 467089069642660275408t^{14} + 103344550287754022100t^{13} + 18522489169628911672t^{12} + 3057026968156022001t^{11} + 460331889515380365t^{10} + 52379385985175568t^{9} + 3337043297368725t^{8} - 17375400117549t^{7} - 24191581345218t^{6} - 2389501975473t^{5} - 141396694713t^{4} - 7204543380t^{3} - 367202565t^{2} - 11625363t - 118098 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: Fl(1, 3, 6)

|λ| = 6.09017 λ ≈ 6.09017; multiplicity 1; Jordan blocks 1 λ ≈ -0.474527 - 5.734048i; multiplicity 1; Jordan blocks 1 λ ≈ -0.474527 + 5.734048i; multiplicity 1; Jordan blocks 1 λ ≈ -5.09017; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 1; Jordan blocks 1 λ ≈ -3.050946; multiplicity 1; Jordan blocks 1 λ ≈ -1.5 - 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ -1.5 + 2.598076i; multiplicity 1; Jordan blocks 1 λ ≈ 3; multiplicity 1; Jordan blocks 1
Sector dimension
9
Spectral radius
$6.09016994$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$6.09017$11
$-0.474527 - 5.734048\mathrm{i}$11
$-0.474527 + 5.734048\mathrm{i}$11
$-5.09017$11
$-5$11
$-3.050946$11
$-1.5 - 2.598076\mathrm{i}$11
$-1.5 + 2.598076\mathrm{i}$11
$3$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda - 3\right) \left(\lambda^{2} - \lambda - 31\right) \left(\lambda^{2} + 3 \lambda + 9\right) \left(\lambda^{3} + 4 \lambda^{2} + 36 \lambda + 101\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right) \left(\lambda - 3\right) \left(\lambda^{2} - \lambda - 31\right) \left(\lambda^{2} + 3 \lambda + 9\right) \left(\lambda^{3} + 4 \lambda^{2} + 36 \lambda + 101\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 0 & 0 & 3 & 0 & 0 & 8 & 0 \\ 3 & 0 & 2 & 0 & 0 & -4 & 0 & 0 & 0 \\ 1 & 0 & -2 & 0 & 0 & 4 & 0 & 3 & 8 \\ 0 & 1 & 3 & -2 & 0 & 2 & 4 & 0 & 0 \\ 0 & 3 & 0 & 2 & 0 & 0 & -4 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & -2 & 0 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0 & 3 & -2 & 0 & 0 \\ 0 & 0 & 0 & 3 & 4 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 4 & 0\end{pmatrix}\notag\end{equation}$$