Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-188-44-g

·

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

Defining data
Ambient
ℙ4P4 × Gr(2, 6)
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2node 3: fundamental weight ω_3node 4: fundamental weight ω_4node 5: fundamental weight ω_5node 6: fundamental weight ω_6 (marked)node 7: fundamental weight ω_7node 8: fundamental weight ω_8node 9: fundamental weight ω_9ω1ω2ω3ω4ω5ω6ω7ω8ω9
Bundle
E^(w6) + E^(2w5) + E^(w1 + w9)
Bundle
O_2(1) + Sym^2 U_2^v + O_1(1) box Q_2
ZL62 label
4051.5Dw7
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(5;6) \\ \det\mathcal{E} &\cong \mathcal{O}(4;5) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 44
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 188
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 188
$\mathrm{c}_1^2\mathrm{c}_2$ = 140
$\mathrm{c}_2^2$ = 104
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 16
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 8 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 + 4 0 x0 + 4 x2
$\mathrm{HH}^{ 3 }$ = x1 + 30 0 0 x1 30
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 41 0 0 x1 - x2 - 3 0 44
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 39t + 108t^{2} + 39t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(342)
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( t - 1 \right) \left( 3t + 1 \right) \left( 2t + 1 \right) ^{2} \left( 20t^{2} + 4t + 1 \right) \left( 356t^{3} + 64t^{2} + 5t - 1 \right) \left( 230564061056t^{11} + 366435334400t^{10} + 259363086592t^{9} + 82844391552t^{8} + 1433227848t^{7} - 6231253224t^{6} - 1487165488t^{5} - 101702782t^{4} + 4786560t^{3} + 856611t^{2} + 3717t - 282 \right)D^{6} + \left( 2t + 1 \right) \left( 206843630454558720t^{19} + 365299978251542528t^{18} + 214981578805925888t^{17} - 48853393368311808t^{16} - 147871817235984384t^{15} - 92562506032049664t^{14} - 25590533293350912t^{13} - 457189355331072t^{12} + 2011987182203600t^{11} + 696134429536928t^{10} + 114949279795608t^{9} + 8849913955512t^{8} + 20974811262t^{7} - 16935256598t^{6} + 5214067655t^{5} + 536331355t^{4} - 24532083t^{3} - 4167975t^{2} - 17406t + 846 \right)D^{5} + \left( 3447393840909312000t^{20} + 7711807912722821120t^{19} + 6980567343561449472t^{18} + 2257046795830398976t^{17} - 1141736628008862720t^{16} - 1439854237148630528t^{15} - 569681411015298816t^{14} - 57605336721268608t^{13} + 36680963008500768t^{12} + 17551151120203536t^{11} + 3634622564223600t^{10} + 385880000352952t^{9} + 10233393368692t^{8} - 2910052699330t^{7} - 490485003290t^{6} - 40428423365t^{5} - 1585228228t^{4} + 15446082t^{3} + 4423038t^{2} + 5613t - 564 \right)D^{4} + t \left( 14479054131819110400t^{19} + 32058860704085954560t^{18} + 30074934881305640960t^{17} + 12308832554109257728t^{16} - 1161531316906039296t^{15} - 3472477526162074112t^{14} - 1359088167665753600t^{13} - 74089800516580992t^{12} + 109604942120032736t^{11} + 41632071253349168t^{10} + 6973555017751840t^{9} + 488167724180560t^{8} - 21469496048580t^{7} - 6821188665902t^{6} - 464820998928t^{5} - 3965117701t^{4} + 807457011t^{3} + 25015329t^{2} + 340467t - 948 \right)D^{3} + 2 t \left( 15995907421819207680t^{19} + 35137860282801596416t^{18} + 33611166894222484480t^{17} + 15214835979235286016t^{16} + 738103176815763456t^{15} - 2495433763821288960t^{14} - 1040765576064143232t^{13} - 55113300921158272t^{12} + 76866814822757792t^{11} + 26536859872683512t^{10} + 3815024310736228t^{9} + 171525119888676t^{8} - 23227504106198t^{7} - 3644243029534t^{6} - 137909394732t^{5} + 6228928461t^{4} + 567116661t^{3} + 11037003t^{2} + 87138t - 141 \right)D^{2} + 8 t ^{2} \left( 4343716239545733120t^{18} + 9486116864325260288t^{17} + 9168856985959757312t^{16} + 4354371895779082752t^{15} + 477247135835727360t^{14} - 503097538314597952t^{13} - 231577651079967712t^{12} - 18474620240328288t^{11} + 12729632313066160t^{10} + 4353942022018072t^{9} + 557192258073820t^{8} + 13645965220706t^{7} - 4408616149295t^{6} - 444368400840t^{5} - 770892838t^{4} + 1926655895t^{3} + 92141637t^{2} + 1598391t + 4512 \right)D + 96 t ^{3} \left( 147745450324684800t^{17} + 321378694637880320t^{16} + 312333098369566208t^{15} + 151901010743165952t^{14} + 20977628581915904t^{13} - 14233145661603776t^{12} - 7163491072787104t^{11} - 788257128603808t^{10} + 263687416326416t^{9} + 96210646086576t^{8} + 11312132763744t^{7} + 90946820872t^{6} - 98807250620t^{5} - 5844821964t^{4} + 443068614t^{3} + 58825410t^{2} + 2000043t + 34263 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ4P4 × Gr(2, 6)

|λ| = 12.44287 λ ≈ 12.44287; multiplicity 1; Jordan blocks 1 λ ≈ -3.721437 - 3.84209i; multiplicity 1; Jordan blocks 1 λ ≈ -3.721437 + 3.84209i; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 4i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 4i; multiplicity 1; Jordan blocks 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1; multiplicity 1; Jordan blocks 1 2
Sector dimension
9
Spectral radius
$12.4428745$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.44287$11
$-3.721437 - 3.84209\mathrm{i}$11
$-3.721437 + 3.84209\mathrm{i}$11
$-2 - 4\mathrm{i}$11
$-2 + 4\mathrm{i}$11
$-3$11
$-2$21, 1
$1$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 3\right) \left(\lambda - 1\right) \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{3} - 5 \lambda^{2} - 64 \lambda - 356\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda + 3\right) \left(\lambda - 1\right) \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{3} - 5 \lambda^{2} - 64 \lambda - 356\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 8 & 36 & 12 & 36 & 64 & 112 & 160 \\ 1 & -1 & 2 & 4 & 0 & 8 & 24 & 36 & 64 \\ 1 & 1 & 0 & 8 & 4 & 4 & 12 & 24 & 32 \\ 0 & 1 & 2 & 0 & 0 & 4 & 8 & 12 & 24 \\ 0 & 1 & -1 & 1 & -1 & -2 & -4 & -4 & 0 \\ 0 & 0 & 1 & 1 & 1 & -2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 5 & -3 & 1 & 5 & 8 \\ 0 & 0 & 0 & 0 & -2 & 4 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & \frac{5}{2} & 0\end{pmatrix}\notag\end{equation}$$