Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 3-54-18-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ2P2 × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + 2w2 + 2w4)
Bundle
O_1(1) box O_2(2) box O_3(2)
ZL62 label
12020.x4108
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;3;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;2;2) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 18
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 54
$\chi(X, \mathrm{T}_X)$ = -52

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 54
$\mathrm{c}_1^2\mathrm{c}_2$ = 96
$\mathrm{c}_2^2$ = 171
$\mathrm{c}_1\mathrm{c}_3$ = -18
$\mathrm{c}_4$ = 105
Hodge diamond
1
00
0 3 0
0 0 0 0
0 9 79 9 0
0 0 0 0
0 3 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = 52 0 52 0
$\mathrm{HH}^{ 3 }$ = 0 0 0 0 0
$\mathrm{HH}^{ 4 }$ = 52 0 0 34 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 13t + 26t^{2} + 13t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(701)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(698)

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( 5t + 1 \right) ^{3} \left( 443t^{3} + 164t^{2} + 20t + 1 \right) \left( 2792t^{3} + 272t^{2} - 52t + 1 \right) \left( 3845029779418368t^{13} + 6432031381222080t^{12} + 4817515389713984t^{11} + 2123336357397600t^{10} + 605140426584426t^{9} + 114375039385336t^{8} + 13915370809202t^{7} + 954175680523t^{6} + 16078796236t^{5} - 2378198379t^{4} - 136601066t^{3} - 345863t^{2} + 49698t - 81 \right)D^{6} + 2 \left( 5t + 1 \right) ^{2} \left( 249676778024744960977920t^{20} + 553469087438152533981696t^{19} + 565037132844161129678080t^{18} + 351449755545696783346560t^{17} + 148171220634667604336312t^{16} + 44450265358159324369448t^{15} + 9654459402539949112444t^{14} + 1505115720631663015035t^{13} + 160923442061832586311t^{12} + 10217564802505850689t^{11} + 140552976544893446t^{10} - 33400607268202131t^{9} - 2424713603465580t^{8} - 18816156136991t^{7} + 3835766213635t^{6} + 108632344316t^{5} + 2020870488t^{4} + 209997550t^{3} + 1252255t^{2} - 73980t + 81 \right)D^{5} + \left( 5t + 1 \right) \left( 20806398168728746748160000t^{21} + 48224717957500273784908800t^{20} + 51735866230507351407950336t^{19} + 34023120665618860246521600t^{18} + 15284304050838043463765392t^{17} + 4937038483164800567424256t^{16} + 1172251941416725134893016t^{15} + 204734564940704107180930t^{14} + 25689849013295334536420t^{13} + 2161171419810966540900t^{12} + 96496351662574520265t^{11} - 1150467448886042484t^{10} - 412013681132405586t^{9} - 17391940617831167t^{8} + 287534904499988t^{7} + 42324826709512t^{6} + 751184553858t^{5} - 27388300292t^{4} - 1208269589t^{3} - 5066662t^{2} + 88380t - 81 \right)D^{4} + 2 t \left( 218467180771651840855680000t^{21} + 533066343265328202391296000t^{20} + 604673406772485080441171200t^{19} + 422616614938156377506968192t^{18} + 203043640129976062090156520t^{17} + 70712921780194410650996424t^{16} + 18303989529005907426908204t^{15} + 3542053859077771915784609t^{14} + 505645855115744805290915t^{13} + 50981364523078239928506t^{12} + 3200964786557717711983t^{11} + 60291206258737051292t^{10} - 8987649193479071497t^{9} - 882028798849744831t^{8} - 31910711064215676t^{7} - 172196037470619t^{6} + 17209074437419t^{5} + 205532501188t^{4} - 5064981591t^{3} - 45899922t^{2} + 1708235t + 297 \right)D^{3} + 4 t ^{2} \left( 241354218757253462278656000t^{20} + 574553640745081539343977600t^{19} + 635174600402485420506536320t^{18} + 432078289199202904643028224t^{17} + 201646905219089317214149408t^{16} + 68007964425145479275842784t^{15} + 16966779439748642819161024t^{14} + 3140597445314251505010714t^{13} + 423274987340550042996714t^{12} + 39196001548870501632523t^{11} + 2063320789525327053878t^{10} - 5059259301748992865t^{9} - 10079411852698230383t^{8} - 763337875035982748t^{7} - 24573310645698487t^{6} - 146245469043394t^{5} + 10015954260779t^{4} + 155899618093t^{3} - 326703723t^{2} + 22964685t + 479718 \right)D^{2} + 8 t ^{2} \left( 131080308462991104513408000t^{20} + 306327712766096321020041600t^{19} + 332118245016014978999543680t^{18} + 221287674617007656970219904t^{17} + 100964443235398445806573144t^{16} + 33194062679056252063439672t^{15} + 8036301681131505648724420t^{14} + 1432985572270062484152537t^{13} + 183614929137564053832095t^{12} + 15685225109560826711263t^{11} + 671892193916145400760t^{10} - 20994770464839318085t^{9} - 5116497688791492911t^{8} - 328131628455019498t^{7} - 9338624325373913t^{6} - 36055818236309t^{5} + 3905031921716t^{4} + 64268764395t^{3} + 163329241t^{2} + 11899593t + 68904 \right)D + 48 t ^{2} \left( 8917027786598034320640000t^{20} + 20575994198022089857248000t^{19} + 22008939012859834424195840t^{18} + 14452200043643239231302656t^{17} + 6488364978720045185252816t^{16} + 2093948069769663383117072t^{15} + 495730088115035801126520t^{14} + 85898621444316488674150t^{13} + 10571020231514617547114t^{12} + 842384985739572287018t^{11} + 28772312470437135900t^{10} - 1996559989387413223t^{9} - 321393864221409369t^{8} - 18401193808685168t^{7} - 469467232135029t^{6} - 661152978472t^{5} + 213015605703t^{4} + 3499205526t^{3} + 15933765t^{2} + 586089t + 1620 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ2P2 × ℙ2P2

|λ| = 44.47212 λ ≈ 44.47212; multiplicity 1; Jordan blocks 1 λ ≈ 12.53595; multiplicity 1; Jordan blocks 1 λ ≈ -7.525473 - 5.734048i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -7.525473 + 5.734048i; multiplicity 2; Jordan blocks 1, 1 λ ≈ -5.008069; multiplicity 1; Jordan blocks 1 λ ≈ -5; multiplicity 4; Jordan blocks 1, 3 λ ≈ -4.949054; multiplicity 2; Jordan blocks 1, 1 2 2 4 2
Sector dimension
13
Spectral radius
$44.4721165$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$44.47212$11
$12.53595$11
$-7.525473 - 5.734048\mathrm{i}$21, 1
$-7.525473 + 5.734048\mathrm{i}$21, 1
$-5.008069$11
$-5$41, 3
$-4.949054$21, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 5\right)^{4} \left(\lambda^{3} + 20 \lambda^{2} + 164 \lambda + 443\right)^{2} \left(\lambda^{3} - 52 \lambda^{2} + 272 \lambda + 2792\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 5\right)^{3} \left(\lambda^{3} + 20 \lambda^{2} + 164 \lambda + 443\right) \left(\lambda^{3} - 52 \lambda^{2} + 272 \lambda + 2792\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 16 & 52 & 52 & 270 & 270 & 768 & 411 & 411 & 4640 & 2320 & 4448 & 22560 \\ 1 & -1 & 3 & 3 & 48 & 48 & 76 & 44 & 44 & 744 & 372 & 558 & 3288 \\ 1 & 2 & 0 & 6 & 16 & 24 & 84 & 44 & 48 & 462 & 240 & 483 & 2320 \\ 1 & 2 & 6 & 0 & 24 & 16 & 84 & 48 & 44 & 462 & 222 & 492 & 2320 \\ 0 & 1 & 1 & 0 & -1 & 6 & 3 & 1 & 4 & 80 & 40 & 52 & 348 \\ 0 & 1 & 0 & 1 & 6 & -1 & 3 & 4 & 1 & 80 & 40 & 52 & 348 \\ 0 & 0 & 1 & 1 & 2 & 2 & 0 & 2 & 2 & 16 & 8 & 28 & 144 \\ 0 & 0 & 1 & 0 & 2 & 0 & 6 & -2 & 4 & 24 & 8 & 32 & 126 \\ 0 & 0 & 0 & 1 & 0 & 2 & 6 & 4 & -2 & 24 & 16 & 28 & 126 \\ 0 & 0 & 0 & 0 & 1 & 2 & \frac{1}{2} & -\frac{1}{2} & 1 & 4 & 5 & 3 & 44 \\ 0 & 0 & 0 & 0 & 1 & -1 & 1 & 2 & -1 & 2 & -5 & 3 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 1 & 4 & 2 & 0 & 16 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$