Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-90-24-a

·

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

Defining data
Ambient
ℙ3P3 × Gr(2, 4)
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_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w1 + 2w4)
Bundle
O_1(1) box Sym^2 U_2^v
ZL62 label
3031.x321
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(4;4) \\ \det\mathcal{E} &\cong \mathcal{O}(3;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 24
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 90
$\chi(X, \mathrm{T}_X)$ = -9

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 90
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 136
$\mathrm{c}_1\mathrm{c}_3$ = 36
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 12 0 0
0 0 0 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 + x1 + 9 0 x0 + 9 x1
$\mathrm{HH}^{ 3 }$ = x1 + 17 0 0 x1 + 16 1
$\mathrm{HH}^{ 4 }$ = 25 0 0 1 0 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 19t + 50t^{2} + 19t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(554)
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) ^{3} \left( 8t^{2} + 20t - 1 \right) \left( 20t^{2} + 4t + 1 \right) \left( 104t^{2} + 20t + 1 \right) \left( 3359269145344t^{10} - 1385289747456t^{9} - 598547090336t^{8} - 113864678656t^{7} - 117181440576t^{6} - 19515080164t^{5} - 609614844t^{4} - 168148164t^{3} - 7366419t^{2} - 40824t - 405 \right)D^{6} + \left( t - 1 \right) ^{2} \left( 1173863010149007360t^{17} + 1570567658569990144t^{16} - 1729753198667063296t^{15} - 269934092328976384t^{14} + 82469354667934720t^{13} - 23837593801470976t^{12} + 30251062268197504t^{11} + 16263206921063552t^{10} + 3244659065425248t^{9} + 419140244319200t^{8} + 43003604070752t^{7} + 6057798425960t^{6} + 474511121884t^{5} + 14807191236t^{4} + 949447455t^{3} + 30255264t^{2} + 122067t + 810 \right)D^{5} + \left( t - 1 \right) \left( 9782191751241728000t^{18} + 2194760764178759680t^{17} - 18010905385411119104t^{16} + 5939931013446339584t^{15} + 1562286385150521088t^{14} - 257517580842414720t^{13} + 546477769225991040t^{12} + 1877484853798560t^{11} - 45291104917555728t^{10} - 5833117163209856t^{9} - 682771835559400t^{8} - 80594346723340t^{7} - 8781239275328t^{6} - 301845085186t^{5} - 13553570307t^{4} + 609298020t^{3} + 26085870t^{2} + 115722t + 405 \right)D^{4} + t \left( 41085205355215257600t^{18} - 35533877455451586560t^{17} - 56771176387445370880t^{16} + 63743010985033807872t^{15} - 8580661945382489600t^{14} - 3760452316229997056t^{13} + 2199630478045722880t^{12} - 1660645042277880800t^{11} - 233599590388349520t^{10} + 76638093343516240t^{9} + 4050066815971752t^{8} + 783953566913560t^{7} + 147240549423280t^{6} + 14435330777364t^{5} + 622906240235t^{4} + 12025801404t^{3} + 199627770t^{2} + 1989000t + 459 \right)D^{3} + 2 t ^{2} \left( 45389369725761617920t^{17} - 42355681034057056256t^{16} - 43249745760567114752t^{15} + 47530700632728567296t^{14} - 4499392098698119040t^{13} - 1879655457240347840t^{12} + 1848026896157214912t^{11} - 1290207782262763616t^{10} - 262499862553662992t^{9} + 37777791920638040t^{8} - 1171837206511440t^{7} + 233457153632904t^{6} + 74238953423878t^{5} + 4693134460821t^{4} + 208491986700t^{3} + 4387302960t^{2} + 106909890t + 758115 \right)D^{2} + 4 t ^{2} \left( 24651123213129154560t^{17} - 24236993651510706176t^{16} - 16920313129102889984t^{15} + 18794433285592092160t^{14} - 1237507248868711296t^{13} - 306197078131796288t^{12} + 852095826642583968t^{11} - 512940123531732272t^{10} - 137503959128613520t^{9} + 8807503549166992t^{8} - 1458929305699664t^{7} - 5698360124608t^{6} + 17476628942806t^{5} + 683412689685t^{4} + 40630735779t^{3} + 1003946067t^{2} + 26597241t + 145800 \right)D + 16 t ^{2} \left( 2515420736033587200t^{17} - 2558196646021201920t^{16} - 1323300479610264576t^{15} + 1517488418403323904t^{14} - 72224378727980032t^{13} + 11440320426037824t^{12} + 78957411074970608t^{11} - 41177618241522168t^{10} - 13334778260005368t^{9} + 348967576610164t^{8} - 166827338462588t^{7} - 6766998980016t^{6} + 672603543174t^{5} + 10666828839t^{4} + 1782552843t^{3} + 55402839t^{2} + 1334853t + 5670 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ3P3 × Gr(2, 4)

|λ| = 20.3923 λ ≈ 20.3923; multiplicity 1; Jordan blocks 1 λ ≈ -10 - 2i; multiplicity 1; Jordan blocks 1 λ ≈ -10 + 2i; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 4i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 4i; multiplicity 1; Jordan blocks 1 λ ≈ 1; multiplicity 3; Jordan blocks 1, 2 λ ≈ -0.3923048; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$20.3923048$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$20.3923$11
$-10 - 2\mathrm{i}$11
$-10 + 2\mathrm{i}$11
$-2 - 4\mathrm{i}$11
$-2 + 4\mathrm{i}$11
$1$31, 2
$-0.3923048$11

Characteristic polynomial

$$\begin{equation}\left(\lambda - 1\right)^{3} \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{2} + 20 \lambda + 104\right) \left(\lambda^{2} - 20 \lambda - 8\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda - 1\right)^{2} \left(\lambda^{2} + 4 \lambda + 20\right) \left(\lambda^{2} + 20 \lambda + 104\right) \left(\lambda^{2} - 20 \lambda - 8\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 16 & 96 & 84 & 75 & 612 & 292 & 780 \\ 1 & -1 & 0 & 32 & 40 & 18 & 189 & 69 & 240 \\ 1 & 3 & 0 & 12 & 0 & 18 & 54 & 42 & 132 \\ 0 & 1 & \frac{3}{2} & \frac{1}{2} & 1 & -3 & 6 & 2 & 9 \\ 0 & 1 & -\frac{1}{4} & -\frac{3}{4} & -\frac{3}{2} & \frac{9}{2} & 18 & 12 & 27 \\ 0 & 0 & 1 & 3 & 6 & 2 & 36 & 12 & 36 \\ 0 & 0 & 0 & \frac{2}{3} & -1 & \frac{19}{12} & \frac{3}{4} & \frac{7}{4} & 1 \\ 0 & 0 & 0 & 4 & 5 & -\frac{1}{4} & -\frac{3}{4} & -\frac{7}{4} & 15 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & \frac{5}{3} & 0\end{pmatrix}\notag\end{equation}$$