Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-288-63-a

·

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

Defining data
Ambient
ℙ2P2 × Gr(2, 5)
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_5node 6: fundamental weight ω_6ω1ω2ω3ω4ω5ω6
Bundle
2 copies of E^(w4) + E^(w1 + w3)
Bundle
2 copies of O_2(1) + O_1(1) box U_2^v
ZL62 label
2041.33h
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(3;5) \\ \det\mathcal{E} &\cong \mathcal{O}(2;3) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 63
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 288
$\chi(X, \mathrm{T}_X)$ = 2

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 288
$\mathrm{c}_1^2\mathrm{c}_2$ = 168
$\mathrm{c}_2^2$ = 98
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 10
Hodge diamond
1
00
0 2 0
0 0 0 0
0 0 4 0 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 + x1 - 2 0 x0 - 2 x1
$\mathrm{HH}^{ 3 }$ = x1 + 52 0 0 x1 - 22 74
$\mathrm{HH}^{ 4 }$ = 63 0 0 0 0 63
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 58t + 170t^{2} + 58t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(195)
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( 2t + 1 \right) \left( 9731t^{9} - 2809t^{8} + 1930t^{7} + 3621t^{6} + 1515t^{5} - 64t^{4} + 416t^{3} + 46t^{2} - 2t - 1 \right) \left( 83989557042996t^{17} - 3153169026458088t^{16} - 10688202318697350t^{15} - 11148427670965685t^{14} - 5061868984488846t^{13} - 2931495709999175t^{12} - 2565826768801700t^{11} - 1049672429015799t^{10} + 105362760566347t^{9} + 5677732497729t^{8} - 86038265873317t^{7} - 43588220217707t^{6} - 3273480672284t^{5} - 80156379318t^{4} + 23422614208t^{3} - 6462578464t^{2} - 44633088t - 1161216 \right)D^{6} + \left( 34326699942586551192t^{27} - 1344200829226397467116t^{26} - 5016611417496139623744t^{25} - 6082010289675780758754t^{24} - 3834837114558219991530t^{23} - 3312120754712547455094t^{22} - 3437566570817733543018t^{21} - 2106858535961349007908t^{20} - 818882526213643147704t^{19} - 562522071383740327305t^{18} - 392500791170875613045t^{17} - 122484046940909007097t^{16} - 12963690363170030818t^{15} - 16276270994151613441t^{14} - 15830398221176558325t^{13} - 6462791775107897591t^{12} - 1648991421215108246t^{11} - 472927431556650144t^{10} - 106309400245879593t^{9} - 10976845572155370t^{8} - 1805612910720261t^{7} - 372211287701856t^{6} - 47342892897268t^{5} - 1689508873796t^{4} + 101841145280t^{3} - 25964407936t^{2} - 124609536t - 2322432 \right)D^{5} + \left( 286055832854887926600t^{27} - 11619998722122636751620t^{26} - 44338495472158428781968t^{25} - 54417035967367292161958t^{24} - 32784997664827461229971t^{23} - 24225711887122883079139t^{22} - 23387237442276190827415t^{21} - 13095316741461926074622t^{20} - 3539400430201006592330t^{19} - 2766910820866975261631t^{18} - 2379313216611997774411t^{17} - 793251586962275751239t^{16} - 92371166716870779906t^{15} - 120116271454936574378t^{14} - 111466504946752931106t^{13} - 40437749211849378575t^{12} - 6965411843381658295t^{11} - 893452999574282967t^{10} - 84512829340847423t^{9} - 10409729913914267t^{8} + 502664148825553t^{7} + 558539779230681t^{6} - 35165034708424t^{5} - 490190932738t^{4} - 186113701696t^{3} + 26006520992t^{2} + 183905280t + 1161216 \right)D^{4} + t \left( 1201434497990529291720t^{26} - 50184467612193845291460t^{25} - 194618289232085425773120t^{24} - 241091630153640639172194t^{23} - 143096541502338640779750t^{22} - 100624065431801383460922t^{21} - 97450627971927546916488t^{20} - 53376209334919659215082t^{19} - 9221436317570463478644t^{18} - 7730363934245446734429t^{17} - 8857651607085637737731t^{16} - 3610510531794755554165t^{15} - 642165279060010052750t^{14} - 510315456248498429401t^{13} - 391620505425581928325t^{12} - 123385012076297916931t^{11} - 17179215503764404766t^{10} - 842877050017711076t^{9} + 166343129278238337t^{8} - 27076871084810568t^{7} - 5301299610456683t^{6} - 695278062728012t^{5} - 162625152669220t^{4} - 3452102663752t^{3} - 91083992832t^{2} - 1253338688t - 193536 \right)D^{3} + 2 t ^{2} \left( 1327299064446679979424t^{25} - 56608716380242637850096t^{24} - 222126565123069608809616t^{23} - 277099430867479708930039t^{22} - 164336009232062745405093t^{21} - 116613613079927756941427t^{20} - 117736396044668913174974t^{19} - 65561898871092099613042t^{18} - 7655872825743768432862t^{17} - 6072078381960850855018t^{16} - 9571904361566940307346t^{15} - 4632724133256459848718t^{14} - 923711134898006659599t^{13} - 513656752343667153967t^{12} - 341766491383013110734t^{11} - 95575325248434726811t^{10} - 11293312812587646975t^{9} - 31423553721363712t^{8} + 166986602596438705t^{7} - 34780538873430640t^{6} - 5517749635856117t^{5} - 605779742918270t^{4} - 81359470615242t^{3} - 1421057816692t^{2} - 51056924064t - 498052224 \right)D^{2} + 4 t ^{2} \left( 720860698794317575032t^{25} - 31208784073560784054716t^{24} - 123477549512602703567184t^{23} - 154816828401393730458555t^{22} - 92206239115329202629951t^{21} - 67299249799457801445855t^{20} - 71043436766553540805830t^{19} - 40480751898492374272110t^{18} - 3654031363085701412688t^{17} - 2511213422189443040700t^{16} - 5347237318317119142314t^{15} - 2900239057179655342377t^{14} - 586170623057276144082t^{13} - 247162097047320410905t^{12} - 146533544201718700731t^{11} - 37343850127518336978t^{10} - 3988820386751490783t^{9} + 80244150055550948t^{8} + 45269364017726321t^{7} - 19561220582128156t^{6} - 2558435289870578t^{5} - 242497389501504t^{4} - 21462562755232t^{3} - 397267556608t^{2} - 15189202944t - 105283584 \right)D + 24 t ^{2} \left( 49038142775123644560t^{25} - 2144389655271277377480t^{24} - 8530524103207786479648t^{23} - 10731732560111078595414t^{22} - 6421473159152924330670t^{21} - 4811323778245704251376t^{20} - 5248201636137253566150t^{19} - 3042271654530473190913t^{18} - 238047610849037548473t^{17} - 139954367133448221102t^{16} - 377861063883546475317t^{15} - 218899103274592932111t^{14} - 43764326133769408632t^{13} - 15085217022028926080t^{12} - 8155354800393918146t^{11} - 1936568811451057580t^{10} - 196301123165903501t^{9} + 5698529660984304t^{8} + 826892192633212t^{7} - 1289114051667028t^{6} - 147700794796458t^{5} - 12477402518304t^{4} - 783616951712t^{3} - 18118340096t^{2} - 633378816t - 3096576 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ2P2 × Gr(2, 5)

|λ| = 8.8103 λ ≈ 8.8103; multiplicity 1; Jordan blocks 1 λ ≈ -5.589145 - 4.418565i; multiplicity 1; Jordan blocks 1 λ ≈ -5.589145 + 4.418565i; multiplicity 1; Jordan blocks 1 λ ≈ 0.9432073 - 2.095592i; multiplicity 1; Jordan blocks 1 λ ≈ 0.9432073 + 2.095592i; multiplicity 1; Jordan blocks 1 λ ≈ -2; multiplicity 1; Jordan blocks 1 λ ≈ -1.470419 - 0.9012513i; multiplicity 1; Jordan blocks 1 λ ≈ -1.470419 + 0.9012513i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7112067 - 0.9377349i; multiplicity 1; Jordan blocks 1 λ ≈ 0.7112067 + 0.9377349i; multiplicity 1; Jordan blocks 1
Sector dimension
10
Spectral radius
$8.8103001$
Diagonalizable at 1
Yes
Simple spectrum at 1
Yes
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$8.8103$11
$-5.589145 - 4.418565\mathrm{i}$11
$-5.589145 + 4.418565\mathrm{i}$11
$0.9432073 - 2.095592\mathrm{i}$11
$0.9432073 + 2.095592\mathrm{i}$11
$-2$11
$-1.470419 - 0.9012513\mathrm{i}$11
$-1.470419 + 0.9012513\mathrm{i}$11
$0.7112067 - 0.9377349\mathrm{i}$11
$0.7112067 + 0.9377349\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{9} + 2 \lambda^{8} - 46 \lambda^{7} - 416 \lambda^{6} + 64 \lambda^{5} - 1515 \lambda^{4} - 3621 \lambda^{3} - 1930 \lambda^{2} + 2809 \lambda - 9731\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 2\right) \left(\lambda^{9} + 2 \lambda^{8} - 46 \lambda^{7} - 416 \lambda^{6} + 64 \lambda^{5} - 1515 \lambda^{4} - 3621 \lambda^{3} - 1930 \lambda^{2} + 2809 \lambda - 9731\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 0 & 4 & 9 & 3 & 3 & 6 & 0 & 0 & 10 \\ 1 & -1 & 0 & 4 & 0 & 2 & 4 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 2 & 2 & 6 & 9 & 0 \\ 0 & 2 & 1 & -1 & 0 & 0 & 0 & 2 & 2 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 & 0 & 2 & 4 & 0 \\ 0 & 0 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 & -4 & 9 & 12 & 3 & 6 & -8 \\ 0 & 0 & 0 & 4 & 4 & -4 & -5 & -2 & -4 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{2} & \frac{7}{2} & 0\end{pmatrix}\notag\end{equation}$$