Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-50-17-b

·

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

Defining data
Ambient
ℙ1P1 × Gr(2, 5)
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
E^(w3) + E^(2w3) + E^(w1 + w3)
Bundle
O_2(1) + O_2(2) + O_1(1) box O_2(1)
ZL62 label
141.27a
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 17
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 50
$\chi(X, \mathrm{T}_X)$ = -38

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 50
$\mathrm{c}_1^2\mathrm{c}_2$ = 92
$\mathrm{c}_2^2$ = 154
$\mathrm{c}_1\mathrm{c}_3$ = -8
$\mathrm{c}_4$ = 52
Hodge diamond
1
00
0 2 0
0 0 0 0
0 2 42 2 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 + x2 + 38 0 x0 + 38 x2
$\mathrm{HH}^{ 3 }$ = x1 + 1 0 0 x1 1
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 15 0 0 x1 - x2 - 2 0 17
$\mathrm{HH}^{ 5 }$ = 1 0 0 1 0
$\mathrm{HH}^{ 6 }$ = 0 0 0 0
$\mathrm{HH}^{ 7 }$ = 0 0 0
$\mathrm{HH}^{ 8 }$ = 0 0

Hilbert series

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 12t + 24t^{2} + 12t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(717)
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( 6t + 1 \right) ^{2} \left( 7t + 1 \right) ^{3} \left( 54224t^{4} + 13248t^{3} + 360t^{2} - 64t + 1 \right) \left( 132881828352t^{9} - 254553064192t^{8} - 226263037440t^{7} - 70238006016t^{6} - 10548276160t^{5} - 744738752t^{4} - 11972576t^{3} + 1094048t^{2} + 34274t - 45 \right)D^{6} + \left( 6t + 1 \right) \left( 7t + 1 \right) ^{2} \left( 6355148917812903936t^{15} - 10272011959770021888t^{14} - 16657270413443960832t^{13} - 9212559154043019264t^{12} - 2754413405564021760t^{11} - 497063050444634112t^{10} - 54568185477271296t^{9} - 3287210785779200t^{8} - 54848958096704t^{7} + 5112014614528t^{6} + 275663879696t^{5} + 2426825504t^{4} - 67331940t^{3} - 4054024t^{2} - 100617t + 90 \right)D^{5} + \left( 7t + 1 \right) \left( 2224302121234516377600t^{17} - 3359665018098754928640t^{16} - 6882306140790310035456t^{15} - 4675818304822406475776t^{14} - 1764188528423582464000t^{13} - 422111873359867003904t^{12} - 66975354185732683264t^{11} - 7013102469239057152t^{10} - 451645659493824896t^{9} - 13098737673139648t^{8} + 327444065773152t^{7} + 39813188621616t^{6} + 1041355820024t^{5} - 1899309036t^{4} - 344556434t^{3} + 2422887t^{2} + 62902t - 45 \right)D^{4} + t \left( 65394482364294781501440t^{17} - 99894888013085830103040t^{16} - 216322492205631435939840t^{15} - 155505311153532172922880t^{14} - 62688433740677542189056t^{13} - 16254465093515485381632t^{12} - 2853370168921460332032t^{11} - 341849568170905926912t^{10} - 26947326955738774144t^{9} - 1209802808273356480t^{8} - 8797387261457888t^{7} + 2303698531324912t^{6} + 132551878891160t^{5} + 3016066324404t^{4} + 21738385806t^{3} - 56773051t^{2} + 2578506t + 441 \right)D^{3} + 2 t ^{2} \left( 72245332897697091944448t^{16} - 119203628456145841864704t^{15} - 237979300879768036270080t^{14} - 162257467069151882444800t^{13} - 62111877695757110065152t^{12} - 15243050314092606867456t^{11} - 2516026731766375143936t^{10} - 280059563036839409920t^{9} - 19984588320068977408t^{8} - 740389976133121472t^{7} + 5929322824856928t^{6} + 2067114284875040t^{5} + 95820205440360t^{4} + 1911898855696t^{3} + 12937782570t^{2} - 4693489t + 716690 \right)D^{2} + 8 t ^{2} \left( 19618344709288434450432t^{16} - 34129809456833239252992t^{15} - 64299781285863553253376t^{14} - 42122466690220700141568t^{13} - 15492079270725478815744t^{12} - 3640237462343750317056t^{11} - 571582441546831987968t^{10} - 59805217149746688000t^{9} - 3902554865134163712t^{8} - 117216625315388800t^{7} + 3185973786760272t^{6} + 455187389809752t^{5} + 18074887605420t^{4} + 319266911100t^{3} + 1921946356t^{2} + 3216395t + 52500 \right)D + 288 t ^{2} \left( 222430212123451637760t^{16} - 400439508104385576960t^{15} - 725963303527353163776t^{14} - 462558147694032001024t^{13} - 165349602657164771328t^{12} - 37641043857343455744t^{11} - 5692888340875323904t^{10} - 567578193701287680t^{9} - 34370039628489536t^{8} - 828416552362240t^{7} + 42457174010112t^{6} + 4346549968936t^{5} + 154395208904t^{4} + 2471839614t^{3} + 13546776t^{2} + 51270t + 225 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 5)

|λ| = 51.68144 λ ≈ 51.68144; multiplicity 1; Jordan blocks 1 λ ≈ 25.03992; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 3; Jordan blocks 1, 2 λ ≈ -6.36068 - 1.201132i; multiplicity 1; Jordan blocks 1 λ ≈ -6.36068 + 1.201132i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 2; Jordan blocks 1, 1 3 2
Sector dimension
9
Spectral radius
$51.6814425$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$51.68144$11
$25.03992$11
$-7$31, 2
$-6.36068 - 1.201132\mathrm{i}$11
$-6.36068 + 1.201132\mathrm{i}$11
$-6$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 6\right)^{2} \left(\lambda + 7\right)^{3} \left(\lambda^{4} - 64 \lambda^{3} + 360 \lambda^{2} + 13248 \lambda + 54224\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 6\right) \left(\lambda + 7\right)^{2} \left(\lambda^{4} - 64 \lambda^{3} + 360 \lambda^{2} + 13248 \lambda + 54224\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 168 & 468 & 1692 & 2556 & 5856 & 20832 & 69120 \\ 1 & -1 & 6 & 156 & 108 & 168 & 1512 & 2376 & 14976 \\ 1 & 1 & 10 & 32 & 164 & 248 & 456 & 1968 & 5856 \\ 0 & 1 & 1 & 9 & 6 & 10 & 152 & 200 & 1512 \\ 0 & 0 & 1 & 1 & -2 & 6 & 8 & 56 & 144 \\ 0 & 0 & 1 & 1 & 8 & 6 & 16 & 88 & 216 \\ 0 & 0 & 0 & \frac{5}{2} & 1 & \frac{3}{2} & 9 & 16 & 156 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 1 & 0 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0\end{pmatrix}\notag\end{equation}$$