Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-102-27-a

·

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

Defining data
Ambient
ℙ2P2 × ℙ2P2 × ℙ2P2
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_5 (marked)node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
E^(w3 + 2w5) + E^(2w1 + w3)
Bundle
O_2(1) box O_3(2) + O_1(2) box O_2(1)
ZL62 label
202020.x3A1x317
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 27
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 102
$\chi(X, \mathrm{T}_X)$ = -10

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 102
$\mathrm{c}_1^2\mathrm{c}_2$ = 108
$\mathrm{c}_2^2$ = 123
$\mathrm{c}_1\mathrm{c}_3$ = 42
$\mathrm{c}_4$ = 21
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 13 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 + 10 0 x0 + 10 0
$\mathrm{HH}^{ 3 }$ = 18 0 0 18 0
$\mathrm{HH}^{ 4 }$ = 28 0 0 1 0 27
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 22t + 56t^{2} + 22t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(613)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(630)

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( 22t - 1 \right) \left( 5t + 1 \right) ^{3} \left( 9t^{2} - 3t + 1 \right) \left( 52t^{2} + 10t + 1 \right) \left( 659791574688t^{10} + 937441026600t^{9} + 763280870804t^{8} + 257005102084t^{7} + 29652101088t^{6} + 27153945t^{5} + 26349037t^{4} + 34067246t^{3} + 1466970t^{2} - 14275t + 85 \right)D^{6} + 2 \left( 5t + 1 \right) ^{2} \left( 1069931213345554560t^{17} + 1842034786351274304t^{16} + 1754113859154883152t^{15} + 852567832943270208t^{14} + 229977943188113760t^{13} + 51057576202478074t^{12} + 13077765770113868t^{11} + 2407813763210250t^{10} + 192062657673677t^{9} + 4437112460110t^{8} + 2240322955265t^{7} + 504571278865t^{6} + 30163853343t^{5} + 271967375t^{4} + 57563215t^{3} + 3133305t^{2} - 21880t + 85 \right)D^{5} + \left( 5t + 1 \right) \left( 89160934445462880000t^{18} + 168051929787090172800t^{17} + 173114926801358425632t^{16} + 94027163319961044000t^{15} + 28139771038321366776t^{14} + 5383488380218345836t^{13} + 911404667311983412t^{12} + 149735330692954812t^{11} + 15513724602103185t^{10} + 766600077732279t^{9} + 126826623752569t^{8} + 30648548257094t^{7} + 2870168167044t^{6} + 112911437032t^{5} + 969903232t^{4} - 200183634t^{3} - 6263045t^{2} + 19365t - 85 \right)D^{4} + 4 t \left( 468094905838680120000t^{18} + 962348594195948976000t^{17} + 1068536925166258373400t^{16} + 648245636529132036624t^{15} + 223076094598424098356t^{14} + 46371173329881561823t^{13} + 6644468417547305748t^{12} + 854746434118565926t^{11} + 106795423876328530t^{10} + 10874958891889155t^{9} + 1365873008051922t^{8} + 244776636945013t^{7} + 28821627535745t^{6} + 1639320856546t^{5} + 34138785341t^{4} - 65714839t^{3} + 837565t^{2} + 217000t + 65 \right)D^{3} + 4 t ^{2} \left( 1034266839567369408000t^{17} + 2103437981863079110800t^{16} + 2334411174524853522960t^{15} + 1387653257718455752008t^{14} + 455909624553155426496t^{13} + 86510035858867038410t^{12} + 10403530057977510944t^{11} + 1118372903835573977t^{10} + 149885026285633019t^{9} + 19241252147567375t^{8} + 2520767833212319t^{7} + 348216557011205t^{6} + 32188136127161t^{5} + 1447764005863t^{4} + 21041640533t^{3} - 74419242t^{2} + 10114280t + 139060 \right)D^{2} + 32 t ^{2} \left( 140428471751604036000t^{17} + 283318150504348716300t^{16} + 314439774105961490400t^{15} + 184013618355970213164t^{14} + 58331868699201337602t^{13} + 10333934082425852856t^{12} + 1093284286357664034t^{11} + 107250242204995975t^{10} + 16633475384294073t^{9} + 2401842833473485t^{8} + 287567924349025t^{7} + 31779515341861t^{6} + 2419719527043t^{5} + 89905583739t^{4} + 940258629t^{3} - 1843320t^{2} + 896890t + 5780 \right)D + 16 t ^{2} \left( 114635487144166560000t^{17} + 230023813180357908000t^{16} + 255339271975163711040t^{15} + 147799459815456431304t^{14} + 45674787020952855960t^{13} + 7706252684251029960t^{12} + 745966629992287440t^{11} + 71132390881138407t^{10} + 12557575047133356t^{9} + 1870677749125185t^{8} + 202139264075333t^{7} + 18730430292867t^{6} + 1221626225077t^{5} + 39210896983t^{4} + 327938695t^{3} + 3726890t^{2} + 447915t + 1700 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ2P2

|λ| = 22 λ ≈ 22; multiplicity 1; Jordan blocks 1 λ ≈ -5 - 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ -5 + 5.196152i; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 3; Jordan blocks 1, 2 λ ≈ -3; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 - 2.598076i; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1.5 + 2.598076i; multiplicity 2; Jordan blocks 1, 1 3 2 2 2
Sector dimension
12
Spectral radius
$22$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$22$11
$-5 - 5.196152\mathrm{i}$11
$-5 + 5.196152\mathrm{i}$11
$-5$31, 2
$-3$21, 1
$1.5 - 2.598076\mathrm{i}$21, 1
$1.5 + 2.598076\mathrm{i}$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right)^{2} \left(\lambda + 5\right)^{3} \left(\lambda - 22\right) \left(\lambda^{2} + 10 \lambda + 52\right) \left(\lambda^{2} - 3 \lambda + 9\right)^{2}\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda + 5\right)^{2} \left(\lambda - 22\right) \left(\lambda^{2} + 10 \lambda + 52\right) \left(\lambda^{2} - 3 \lambda + 9\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 12 & 16 & 12 & 78 & 48 & 27 & 78 & 400 & 224 & 144 & 600 \\ 1 & 0 & 2 & 0 & 16 & 12 & 4 & 16 & 78 & 48 & 24 & 112 \\ 1 & 3 & -1 & 3 & 8 & 0 & 4 & 8 & 48 & 24 & 18 & 88 \\ 1 & 0 & 2 & 0 & 16 & 12 & 4 & 16 & 78 & 60 & 27 & 112 \\ 0 & 1 & 3 & -\frac{1}{2} & 5 & 3 & 3 & 5 & 0 & 8 & -2 & 9 \\ 0 & 1 & 0 & 1 & 2 & 0 & 0 & 2 & 16 & 16 & 4 & 24 \\ 0 & 1 & -4 & 1 & -10 & 0 & -6 & -10 & 32 & 0 & 16 & 36 \\ 0 & 0 & 1 & \frac{3}{2} & 0 & 3 & 0 & 0 & 16 & 8 & 6 & 27 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 0 & 4 & 6 & 0 & 3 & 12 \\ 0 & 0 & 0 & 0 & \frac{3}{2} & -\frac{1}{4} & \frac{1}{2} & 0 & \frac{5}{2} & -1 & \frac{3}{2} & 2 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -6 & -10 & 4 & -6 & -8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 2 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$