Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-56-18-b

·

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

Defining data
Ambient
ℙ2P2 × 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_4 (marked)node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(2w4) + E^(2w1 + w4)
Bundle
O_2(2) + O_1(2) box O_2(1)
ZL62 label
2031.8x327
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;4) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 56
$\chi(X, \mathrm{T}_X)$ = -30

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 56
$\mathrm{c}_1^2\mathrm{c}_2$ = 92
$\mathrm{c}_2^2$ = 148
$\mathrm{c}_1\mathrm{c}_3$ = 4
$\mathrm{c}_4$ = 40
Hodge diamond
1
00
0 2 0
0 2 2 0
0 3 36 3 0
0 2 2 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 + 30 0 x0 + 30 0
$\mathrm{HH}^{ 3 }$ = 21 0 0 21 0
$\mathrm{HH}^{ 4 }$ = 30 0 0 12 0 18
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 28t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(693)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(690)

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( 42t - 1 \right) \left( 2t + 1 \right) ^{2} \left( 6t + 1 \right) ^{4} \left( 68t^{2} + 4t + 1 \right) \left( 319019902054272t^{11} + 555310260595008t^{10} + 340912700329824t^{9} + 108491441250384t^{8} + 21419633988504t^{7} + 3003204240684t^{6} + 323672338602t^{5} + 25578099999t^{4} + 1270139859t^{3} + 31675439t^{2} + 204357t + 154 \right)D^{6} + \left( 2t + 1 \right) \left( 6t + 1 \right) ^{3} \left( 229602451747284209664t^{16} + 509026174830937737216t^{15} + 442211942117408560128t^{14} + 209630581898670724608t^{13} + 63158417950841519616t^{12} + 13379909066050480128t^{11} + 2146553905492027008t^{10} + 272084107946892480t^{9} + 27403757018818704t^{8} + 2141895436356168t^{7} + 124082211959268t^{6} + 5120832930030t^{5} + 165531272052t^{4} + 5099398491t^{3} + 119670462t^{2} + 611839t + 308 \right)D^{5} + \left( 6t + 1 \right) ^{2} \left( 22960245174728420966400t^{18} + 62069856708456151326720t^{17} + 69122584272731736526848t^{16} + 43075584434902469455872t^{15} + 17167485798671394733056t^{14} + 4755261854884217660928t^{13} + 976887826332997752576t^{12} + 156829628319753166848t^{11} + 20251717857634365504t^{10} + 2089727874690248736t^{9} + 165834865889056080t^{8} + 9549250911468864t^{7} + 367264647549980t^{6} + 7812777509798t^{5} + 2440624855t^{4} - 4410168243t^{3} - 97987427t^{2} - 425555t - 154 \right)D^{4} + t \left( 6t + 1 \right) \left( 578598178403156208353280t^{18} + 1578629519142714301317120t^{17} + 1782718098239424782131200t^{16} + 1139975530838573868675072t^{15} + 472610000435067993581568t^{14} + 137908835260784130336768t^{13} + 30135668629692981376512t^{12} + 5179889391413165452800t^{11} + 720818611382330891136t^{10} + 81077807732431807872t^{9} + 7157323500035316768t^{8} + 474089491502012016t^{7} + 22329100476091032t^{6} + 698491436604552t^{5} + 13188805439034t^{4} + 131881868140t^{3} + 539205449t^{2} - 1499728t + 1133 \right)D^{3} + 2 t ^{2} \left( 3835279353986635438227456t^{18} + 10687597690071785058975744t^{17} + 12415815564918110608932864t^{16} + 8260262030846759229812736t^{15} + 3605338735438979685141504t^{14} + 1119201412389340066066944t^{13} + 262111089225016150053120t^{12} + 48492407661562145983104t^{11} + 7295611231468906485696t^{10} + 896040192684233869920t^{9} + 87945628086841964688t^{8} + 6644701251581992344t^{7} + 369038190148674552t^{6} + 14232935198579460t^{5} + 352610676799906t^{4} + 5111079647015t^{3} + 48749138398t^{2} + 484493962t + 549920 \right)D^{2} + 24 t ^{2} \left( 347158907041893725011968t^{18} + 939838972790753228759040t^{17} + 1049693917356729016418304t^{16} + 669017120361358168065024t^{15} + 279670936065893952567552t^{14} + 83388466847560622150016t^{13} + 18868839876169523853120t^{12} + 3396802853640772304928t^{11} + 499126048620044592528t^{10} + 59646994841921899608t^{9} + 5631948819627499980t^{8} + 402596253729484002t^{7} + 20696833980675558t^{6} + 714874524347709t^{5} + 14876045077976t^{4} + 155653394962t^{3} + 1100014742t^{2} + 17829314t + 38500 \right)D + 432 t ^{2} \left( 7872084059906887188480t^{18} + 20889134099335570636800t^{17} + 22683457682528127123456t^{16} + 14014001356826319608832t^{15} + 5676525945780456165120t^{14} + 1643574539310297496704t^{13} + 362827244790710891712t^{12} + 64061611929287995104t^{11} + 9251082576301455216t^{10} + 1082106852800170824t^{9} + 99074666551198932t^{8} + 6779414528983998t^{7} + 327927022283370t^{6} + 10359448158717t^{5} + 184353173594t^{4} + 1246258243t^{3} + 1451436t^{2} + 106777t + 308 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 4)

|λ| = 42 λ ≈ 42; multiplicity 1; Jordan blocks 1 λ ≈ -2 - 8i; multiplicity 1; Jordan blocks 1 λ ≈ -2 + 8i; multiplicity 1; Jordan blocks 1 λ ≈ -6; multiplicity 4; Jordan blocks 1, 3 λ ≈ -2; multiplicity 2; Jordan blocks 2 4 2
Sector dimension
9
Spectral radius
$42$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$42$11
$-2 - 8\mathrm{i}$11
$-2 + 8\mathrm{i}$11
$-6$41, 3
$-2$22

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 6\right)^{4} \left(\lambda - 42\right) \left(\lambda^{2} + 4 \lambda + 68\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right)^{2} \left(\lambda + 6\right)^{3} \left(\lambda - 42\right) \left(\lambda^{2} + 4 \lambda + 68\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 36 & 72 & 792 & 216 & 504 & 6336 & 4416 & 37440 \\ 1 & 0 & 8 & 72 & 16 & 56 & 600 & 408 & 3840 \\ 1 & 3 & 4 & 84 & 26 & 52 & 732 & 528 & 4416 \\ 0 & 1 & 1 & 4 & 1 & 8 & 52 & 32 & 360 \\ 0 & 1 & 0 & 8 & -2 & 0 & 56 & 40 & 288 \\ 0 & 0 & 2 & 6 & 2 & 0 & 68 & 52 & 432 \\ 0 & 0 & 0 & 2 & 0 & 3 & 6 & 6 & 72 \\ 0 & 0 & 0 & 1 & 1 & -2 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 1 & 0\end{pmatrix}\notag\end{equation}$$