Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-96-27-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ4P4
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_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(3w3) + E^(w1 + w3)
Bundle
O_2(3) + O_1(1) box O_2(1)
ZL62 label
2040.Eh
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 96
$\chi(X, \mathrm{T}_X)$ = -16

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 96
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 99
$\mathrm{c}_1\mathrm{c}_3$ = 30
$\mathrm{c}_4$ = -9
Hodge diamond
1
00
0 2 0
0 5 5 0
0 0 5 0 0
0 5 5 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 + 16 0 x0 + 16 x2
$\mathrm{HH}^{ 3 }$ = x1 + 15 0 0 x1 15
$\mathrm{HH}^{ 4 }$ = x1 - x2 + x3 - 20 0 0 x1 - x2 - 47 x3 27
$\mathrm{HH}^{ 5 }$ = x3 - 9 0 0 x3 - 9 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 50t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(695)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(693)

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( 4t + 1 \right) ^{2} \left( 8t + 1 \right) ^{2} \left( 58349t^{4} + 1959t^{3} - 937t^{2} + 57t - 1 \right) \left( 3403806158864469504t^{13} + 3207544992332134272t^{12} + 1654566289916372256t^{11} + 668163400815900320t^{10} + 191826416269943544t^{9} + 34579591517143000t^{8} + 3681651801575952t^{7} + 218331047370320t^{6} + 6634789431744t^{5} + 122035465580t^{4} + 4461093987t^{3} + 152848307t^{2} - 5310696t + 9792 \right)D^{6} + \left( 4t + 1 \right) \left( 8t + 1 \right) \left( 800790220192366378150428672t^{20} + 1106256738891816572105158656t^{19} + 771045708228751342606629888t^{18} + 381802457275731160101593856t^{17} + 144106498330063511222230720t^{16} + 39681666878655125681298144t^{15} + 7550092914199537022128208t^{14} + 943583607966324738060360t^{13} + 70974433501173956722632t^{12} + 2357293186251592280568t^{11} - 65631628581132033080t^{10} - 9775611316557583392t^{9} - 402987476760604174t^{8} - 4584732041643813t^{7} + 414939785826405t^{6} + 28526190259857t^{5} + 1188407873431t^{4} + 2676168279t^{3} + 679772276t^{2} - 15638328t + 19584 \right)D^{5} + \left( 213544058717964367506780979200t^{22} + 351825900734069634040910315520t^{21} + 289276378552051587526613520384t^{20} + 164779604409613848951567062016t^{19} + 71766939196485329150569351936t^{18} + 23839102761162353522917607808t^{17} + 5877578269698622725373539200t^{16} + 1052322499909240204880026752t^{15} + 134292474255023698330378352t^{14} + 11904678366395970019644120t^{13} + 698709889032457296745984t^{12} + 24269937948179281219776t^{11} + 295894887368540478104t^{10} - 16081242928238025180t^{9} - 1700048446148961318t^{8} - 126583662856868937t^{7} - 5611808196164594t^{6} - 88857776223240t^{5} - 189390032350t^{4} - 34617029190t^{3} - 571629563t^{2} + 9400248t - 9792 \right)D^{4} + t \left( 896885046615450343528480112640t^{21} + 1400927552617242767064646287360t^{20} + 1110521750931917956834494013440t^{19} + 622243290559837433426955921408t^{18} + 267589393969562334633093486080t^{17} + 86826372360473135891234226432t^{16} + 20660236583097063343732288704t^{15} + 3546458844289044296879651232t^{14} + 434508329940587206430013424t^{13} + 37515060401781934247271192t^{12} + 2243112795275043111804920t^{11} + 91742639561375361176088t^{10} + 2687373955375244045352t^{9} + 68334469833645241464t^{8} + 1254881602008596934t^{7} - 42997133081618415t^{6} - 2857021164965985t^{5} + 27163771995795t^{4} + 1581551133649t^{3} - 25510858443t^{2} + 280411794t + 62424 \right)D^{3} + 2 t ^{2} \left( 990844432451354665231463743488t^{20} + 1482826779997676194930828640256t^{19} + 1142650736209358381272776284160t^{18} + 632696175325510698327116084736t^{17} + 268959671285920704839495826560t^{16} + 85325072364592777918035562944t^{15} + 19640029121479769246334890208t^{14} + 3241561383399694218428397552t^{13} + 381687647412913744249950960t^{12} + 31915091143142201192357064t^{11} + 1892011481061220009015072t^{10} + 81147684533899288221720t^{9} + 2739555348268228654820t^{8} + 80914713713726321862t^{7} + 1491760704875833018t^{6} - 31667315264989413t^{5} - 1753431825929297t^{4} + 25336672772817t^{3} + 844109434031t^{2} - 5852389782t + 84069216 \right)D^{2} + 8 t ^{2} \left( 269065513984635103058544033792t^{20} + 389756185486255337429039382528t^{19} + 294091516426052069873124295680t^{18} + 161491052609389063700108813568t^{17} + 67984015072563293425720821184t^{16} + 21151269248734312876090336992t^{15} + 4733095599461359854717275696t^{14} + 755579849990832454419019080t^{13} + 85968001289773786694684048t^{12} + 6980681677331710126806288t^{11} + 408313755573062188720196t^{10} + 17873860644157389111648t^{9} + 641283586496526515322t^{8} + 19719008227432005387t^{7} + 340688513212524396t^{6} - 7142477511278280t^{5} - 303173422276900t^{4} + 5228012865555t^{3} + 113415177222t^{2} - 117857172t + 6746688 \right)D + 192 t ^{2} \left( 4575944115384950732288163840t^{20} + 6480197127808436289170472960t^{19} + 4819381147808747102216360448t^{18} + 2631834639446085538363648896t^{17} + 1099863968509613900676084896t^{16} + 337153727091907284822246336t^{15} + 73846018713597843131710744t^{14} + 11491665852389322669567416t^{13} + 1273064493256929913005040t^{12} + 100968207540899380346680t^{11} + 5830925675892663493468t^{10} + 257510481742272590956t^{9} + 9515642088131963346t^{8} + 296353702147381921t^{7} + 4909113131991732t^{6} - 97102755009744t^{5} - 3302299493408t^{4} + 73632611898t^{3} + 1045409004t^{2} + 4407352t + 45696 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ4P4

|λ| = 30.32195 λ ≈ 30.32195; multiplicity 1; Jordan blocks 1 λ ≈ 16.32053 - 7.506582i; multiplicity 1; Jordan blocks 1 λ ≈ 16.32053 + 7.506582i; multiplicity 1; Jordan blocks 1 λ ≈ -8; multiplicity 2; Jordan blocks 2 λ ≈ -6; multiplicity 1; Jordan blocks 1 λ ≈ -5.963014; multiplicity 1; Jordan blocks 1 λ ≈ -4; multiplicity 2; Jordan blocks 2 2 2
Sector dimension
9
Spectral radius
$30.3219451$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$30.32195$11
$16.32053 - 7.506582\mathrm{i}$11
$16.32053 + 7.506582\mathrm{i}$11
$-8$22
$-6$11
$-5.963014$11
$-4$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 6\right) \left(\lambda + 8\right)^{2} \left(\lambda^{4} - 57 \lambda^{3} + 937 \lambda^{2} - 1959 \lambda - 58349\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 4\right)^{2} \left(\lambda + 6\right) \left(\lambda + 8\right)^{2} \left(\lambda^{4} - 57 \lambda^{3} + 937 \lambda^{2} - 1959 \lambda - 58349\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 2 & 108 & 18 & 18 & 756 & 504 & 432 & 3780 \\ 2 & 0 & 6 & 108 & 0 & 144 & 756 & 0 & 72 \\ 1 & 0 & 9 & 2 & 2 & 108 & 45 & 45 & 432 \\ 0 & 1 & 2 & 9 & 0 & 15 & 108 & 0 & 0 \\ 0 & 2 & 0 & 6 & 0 & 0 & 144 & 108 & 756 \\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 18 \\ 0 & 0 & 0 & 1 & 0 & 3 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 & 1 & -1 & 15 & 9 & 106 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0\end{pmatrix}\notag\end{equation}$$