Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-150-36-a

·

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

Defining data
Ambient
ℙ2P2 × 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_5 (marked)node 6: fundamental weight ω_6node 7: fundamental weight ω_7node 8: fundamental weight ω_8ω1ω2ω3ω4ω5ω6ω7ω8
Bundle
E^(w2 + w4 + w5)
Bundle
Q_1 box Q_2 box O_3(1)
ZL62 label
202040.x4Q00
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;2;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 36
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 150
$\chi(X, \mathrm{T}_X)$ = -4

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 150
$\mathrm{c}_1^2\mathrm{c}_2$ = 120
$\mathrm{c}_2^2$ = 120
$\mathrm{c}_1\mathrm{c}_3$ = 48
$\mathrm{c}_4$ = 18
Hodge diamond
1
00
0 3 0
0 0 0 0
0 0 10 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 + x1 + 4 0 x0 + 4 x1
$\mathrm{HH}^{ 3 }$ = x1 + 23 0 0 x1 + 5 18
$\mathrm{HH}^{ 4 }$ = 36 0 0 0 0 36
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 31t + 86t^{2} + 31t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(346)
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) ^{2} \left( 37t^{3} - 32t^{2} - 2t - 1 \right) \left( 863t^{4} + 762t^{3} + 132t^{2} - t - 1 \right) \left( 120107247433164t^{13} - 227501228827128t^{12} + 192072129023273t^{11} - 16760167703785t^{10} - 27529566608396t^{9} - 3040895358934t^{8} - 1271647440842t^{7} + 880955817294t^{6} + 132813632041t^{5} + 10775328753t^{4} + 3423592272t^{3} + 148038936t^{2} + 775252t + 18900 \right)D^{6} + \left( t - 1 \right) \left( 80538034873555553364t^{21} - 208492223146908938124t^{20} + 206627178491798165429t^{19} - 28249687508021688695t^{18} - 89652294125492520620t^{17} + 30385152699367938298t^{16} + 12119231442320731245t^{15} - 574894159805258007t^{14} - 1017371304360769894t^{13} - 726277323737901769t^{12} + 27094602168635193t^{11} + 46958571462922428t^{10} + 8963889691547163t^{9} + 1670570350768747t^{8} + 297802519637077t^{7} + 41227881351960t^{6} + 3199130359227t^{5} + 154391930386t^{4} + 18279850588t^{3} + 605386448t^{2} + 2325756t + 37800 \right)D^{5} + \left( 671150290612962944700t^{22} - 2254324640513454908160t^{21} + 3180249955235717306575t^{20} - 1886494384961950508313t^{19} - 176824671791137965879t^{18} + 598882624512528841089t^{17} - 85604629414518127585t^{16} - 35384321543219403353t^{15} - 11001747571050096569t^{14} - 2572537383934059776t^{13} + 3590783828159303399t^{12} + 86349032419049563t^{11} - 140176861484241598t^{10} - 15522272188070913t^{9} - 4542830425303686t^{8} - 993108013093517t^{7} - 72188279740318t^{6} - 1678765407672t^{5} - 45960552627t^{4} + 15267326300t^{3} + 546545184t^{2} + 2464256t + 18900 \right)D^{4} + t \left( 2818831220574444367740t^{21} - 8959102708288138295280t^{20} + 12406739903778294877655t^{19} - 7351673639928524078656t^{18} - 193140054874664478467t^{17} + 1634016566193085233538t^{16} - 199616452577370551755t^{15} - 14454661525818885784t^{14} - 58266282266807350093t^{13} - 11684041038218241855t^{12} + 9582045645686499482t^{11} + 329949603023086315t^{10} - 215073743260658919t^{9} + 3461267840373510t^{8} - 9362094797907970t^{7} - 2053485583196855t^{6} - 148113873095225t^{5} - 4497723922181t^{4} - 100791686028t^{3} - 2226776524t^{2} - 35331796t - 12852 \right)D^{3} + 2 t ^{2} \left( 3114137348444148063408t^{20} - 9467397807339104751828t^{19} + 12885057717717292037237t^{18} - 7464114280407790698990t^{17} - 24732255718711312157t^{16} + 1285193400423859697260t^{15} - 133036394722017540671t^{14} + 53965778129054803740t^{13} - 64960860947335239030t^{12} - 10327218564675027785t^{11} + 7105892483126099643t^{10} + 389717498969708327t^{9} - 67801147630350380t^{8} + 23001085240365712t^{7} - 5216079005778824t^{6} - 1158769799960252t^{5} - 67137376496156t^{4} - 1692191698712t^{3} - 45859285594t^{2} - 1325106248t - 13684700 \right)D^{2} + 16 t ^{2} \left( 422824683086166655161t^{20} - 1242631728030760339947t^{19} + 1666492195001262656627t^{18} - 937516103200054869352t^{17} - 472926255781750675t^{16} + 135170148028087617863t^{15} - 12475555054211614749t^{14} + 12249094306367558645t^{13} - 8307111813436280739t^{12} - 1110323186117027451t^{11} + 695606133733814195t^{10} + 53137134160139567t^{9} + 489158022903876t^{8} + 3800768432538486t^{7} - 324447486927790t^{6} - 82509882912743t^{5} - 4134961795004t^{4} - 98554342572t^{3} - 3154245942t^{2} - 90813756t - 686700 \right)D + 96 t ^{2} \left( 28763583883412697630t^{20} - 82565157821092337910t^{19} + 109517623830669017965t^{18} - 60016586275684111418t^{17} - 296695502357108840t^{16} + 7682894711690578152t^{15} - 670488085151481001t^{14} + 1016212528815896622t^{13} - 527815179825662709t^{12} - 63745416020321798t^{11} + 37242579198679159t^{10} + 3495624429041374t^{9} + 311046127286084t^{8} + 268439041372943t^{7} - 7412288305800t^{6} - 3065224971009t^{5} - 144045857670t^{4} - 3670351332t^{3} - 134919666t^{2} - 3354876t - 18900 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × ℙ2P2 × ℙ4P4

|λ| = 13.42172 λ ≈ 13.42172; multiplicity 1; Jordan blocks 1 λ ≈ -6.440035 - 0.4834011i; multiplicity 1; Jordan blocks 1 λ ≈ -6.440035 + 0.4834011i; multiplicity 1; Jordan blocks 1 λ ≈ -1.525473 - 5.734048i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.525473 + 5.734048i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -1.541651; multiplicity 1; Jordan blocks 1 λ ≈ 1.050946; multiplicity 2; Jordan blocks 1, 1 λ ≈ 1; multiplicity 3; Jordan blocks 1, 2 2 2 2 3
Sector dimension
13
Spectral radius
$13.4217203$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$13.42172$11
$-6.440035 - 0.4834011\mathrm{i}$11
$-6.440035 + 0.4834011\mathrm{i}$11
$-1.525473 - 5.734048\mathrm{i}$21, 1
$-1.525473 + 5.734048\mathrm{i}$21, 1
$-1.541651$11
$1.050946$21, 1
$1$31, 2

Characteristic polynomial

$$\begin{equation}\left(\lambda - 1\right)^{3} \left(\lambda^{3} + 2 \lambda^{2} + 32 \lambda - 37\right)^{2} \left(\lambda^{4} + \lambda^{3} - 132 \lambda^{2} - 762 \lambda - 863\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda - 1\right)^{2} \left(\lambda^{3} + 2 \lambda^{2} + 32 \lambda - 37\right) \left(\lambda^{4} + \lambda^{3} - 132 \lambda^{2} - 762 \lambda - 863\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 4 & 12 & 18 & 15 & 18 & 15 & 80 & 40 & 96 & 105 \\ 1 & 0 & 0 & 2 & 2 & -2 & 0 & 6 & 4 & 6 & -3 & 9 & 24 \\ 1 & 0 & 0 & 2 & 2 & 6 & 4 & -2 & 0 & 6 & 6 & 9 & 24 \\ 1 & 0 & 0 & -1 & 2 & 6 & 4 & 6 & 4 & 27 & 18 & 27 & 48 \\ 0 & 1 & 1 & -3 & 0 & 2 & -4 & 2 & -4 & -14 & -6 & -18 & -36 \\ 0 & 1 & 0 & \frac{5}{2} & 0 & -1 & -\frac{1}{2} & 0 & \frac{5}{2} & 7 & 3 & 9 & 18 \\ 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 6 & 6 & 6 & 9 \\ 0 & 0 & 1 & \frac{5}{2} & 0 & 0 & \frac{5}{2} & -1 & -\frac{1}{2} & 7 & 3 & 9 & 18 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 1 & 6 & 2 & 6 & 9 \\ 0 & 0 & 0 & 0 & 4 & 6 & 8 & 0 & -1 & 5 & 0 & 9 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2 & -3 & 2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{5}{3} & -\frac{5}{3} & -\frac{13}{3} & \frac{5}{3} & \frac{2}{3} & -\frac{10}{3} & 0 & -6 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{3} & 1 & 3 & 0\end{pmatrix}\notag\end{equation}$$