Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 2-45-16-b

·

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

Defining data
Ambient
ℙ1P1 × Gr(2, 5)
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_5ω1ω2ω3ω4ω5
Bundle
2 copies of E^(w3) + E^(w1 + 2w3)
Bundle
2 copies of O_2(1) + O_1(1) box O_2(2)
ZL62 label
141.22x311
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;5) \\ \det\mathcal{E} &\cong \mathcal{O}(1;4) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 16
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 45
$\chi(X, \mathrm{T}_X)$ = -50

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 45
$\mathrm{c}_1^2\mathrm{c}_2$ = 90
$\mathrm{c}_2^2$ = 173
$\mathrm{c}_1\mathrm{c}_3$ = -20
$\mathrm{c}_4$ = 94
Hodge diamond
1
00
0 2 0
0 0 0 0
0 8 72 8 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 + x2 + 50 0 x0 + 50 x2
$\mathrm{HH}^{ 3 }$ = x1 0 0 x1 0
$\mathrm{HH}^{ 4 }$ = x1 - x2 + 43 0 0 x1 - x2 + 27 0 16
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 11t + 21t^{2} + 11t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(723)
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( 7t + 1 \right) ^{3} \left( 2145391t^{6} + 1134572t^{5} + 206414t^{4} + 11335t^{3} - 592t^{2} - 52t + 1 \right) \left( 448140825243194t^{13} + 2238511482500884t^{12} + 2491382790960716t^{11} + 1367102190337354t^{10} + 452708298099839t^{9} + 98132641312520t^{8} + 14352296450821t^{7} + 1409362170295t^{6} + 88633263546t^{5} + 3087339545t^{4} + 26269996t^{3} - 1664434t^{2} - 37328t + 36 \right)D^{6} + \left( 7t + 1 \right) ^{2} \left( 141331282101770219171538t^{20} + 808740539236768922172054t^{19} + 1239537028856514120806600t^{18} + 988411437183483889116706t^{17} + 498771824985816924559349t^{16} + 174072992899968734039285t^{15} + 43992303227072971684171t^{14} + 8236460738159609181716t^{13} + 1149905975061355239681t^{12} + 118631462496296689465t^{11} + 8769384114533326628t^{10} + 430047763072158417t^{9} + 10838168458594937t^{8} - 107482778699107t^{7} - 17609934640686t^{6} - 522369340788t^{5} - 5670599330t^{4} + 63963980t^{3} + 5935800t^{2} + 110760t - 72 \right)D^{5} + \left( 7t + 1 \right) \left( 8244324789269929451673050t^{21} + 48720326073220357144406580t^{20} + 76829459885991277732075930t^{19} + 63228682570648840353833028t^{18} + 33089458797656060814948119t^{17} + 12054220511925710541726030t^{16} + 3207178728028266898427744t^{15} + 639531780853607477153296t^{14} + 96664466979839933662909t^{13} + 11066013375510103201962t^{12} + 945951614330480996116t^{11} + 58331737421375391178t^{10} + 2399639464090175174t^{9} + 52177698444858903t^{8} - 156081963655656t^{7} - 31448787943166t^{6} + 131150021186t^{5} + 37868970003t^{4} + 669726356t^{3} - 2514282t^{2} - 69432t + 36 \right)D^{4} + t \left( 242383148804535925879187670t^{21} + 1475459134532865563961008430t^{20} + 2410181813169978796127101090t^{19} + 2063601358855912720606069288t^{18} + 1129034478259423707767689943t^{17} + 432491764941370315463133403t^{16} + 121859622211968942437531842t^{15} + 25963632679951748677785901t^{14} + 4241942533005497003082032t^{13} + 533222632896495002808635t^{12} + 51198403986141398764483t^{11} + 3676139078818390425480t^{10} + 188377391768619765015t^{9} + 6134436306168115164t^{8} + 74627968899479835t^{7} - 3074133598268739t^{6} - 159997908591098t^{5} - 2718178425798t^{4} - 10211156020t^{3} + 41175596t^{2} - 2958064t - 400 \right)D^{3} + 2 t ^{2} \left( 267775669155487308590340664t^{20} + 1637177856417653725132023598t^{19} + 2615676864424365881828575446t^{18} + 2179825720360496561934437838t^{17} + 1157840837807064034824657182t^{16} + 429655609498783576789325114t^{15} + 117006591792003613822408025t^{14} + 24030797290288561337977541t^{13} + 3772169624997319628161223t^{12} + 453619641458978517774025t^{11} + 41416056594650477637602t^{10} + 2800698888733174297096t^{9} + 132653488603002223181t^{8} + 3777558833199864757t^{7} + 21072444232098633t^{6} - 2957800762208757t^{5} - 118637707714392t^{4} - 1881956470548t^{3} - 9937754668t^{2} - 14826460t - 767376 \right)D^{2} + 4 t ^{2} \left( 145429889282721555527512602t^{20} + 892001354248898147641263828t^{19} + 1401323015680774536521726184t^{18} + 1144072417621647046494600150t^{17} + 594077693267157649746375079t^{16} + 215105773901505221407263547t^{15} + 57039349085271582012114337t^{14} + 11378461689443684390234924t^{13} + 1729388411233221357505469t^{12} + 200520258528415463660318t^{11} + 17545789602109881624095t^{10} + 1125874211554068139622t^{9} + 49565169875141904386t^{8} + 1222452745252312311t^{7} - 2580849278617380t^{6} - 1345667386029480t^{5} - 45080983865424t^{4} - 645895517732t^{3} - 3390653660t^{2} - 12163988t - 101232 \right)D + 48 t ^{2} \left( 4946594873561957671003830t^{20} + 30405545319731023576375920t^{19} + 47213698394864976397400460t^{18} + 37999642600615541809592094t^{17} + 19420572416299910712376123t^{16} + 6910254679340812774758973t^{15} + 1797558582396411674987845t^{14} + 351023116078367969867432t^{13} + 52083996309276675812216t^{12} + 5873849610414031616363t^{11} + 497187387059747633828t^{10} + 30576618726079989437t^{9} + 1263852346901080916t^{8} + 26942075142146205t^{7} - 293524446137457t^{6} - 39458109616503t^{5} - 1170200231379t^{4} - 15489000467t^{3} - 80639342t^{2} - 428414t - 1224 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × Gr(2, 5)

|λ| = 57.67502 λ ≈ 57.67502; multiplicity 1; Jordan blocks 1 λ ≈ 20.05211; multiplicity 1; Jordan blocks 1 λ ≈ -7.005767; multiplicity 1; Jordan blocks 1 λ ≈ -7; multiplicity 3; Jordan blocks 3 λ ≈ -6.754115; multiplicity 1; Jordan blocks 1 λ ≈ -5.983622 - 1.844083i; multiplicity 1; Jordan blocks 1 λ ≈ -5.983622 + 1.844083i; multiplicity 1; Jordan blocks 1 3
Sector dimension
9
Spectral radius
$57.6750207$
Diagonalizable at 1
No
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$57.67502$11
$20.05211$11
$-7.005767$11
$-7$33
$-6.754115$11
$-5.983622 - 1.844083\mathrm{i}$11
$-5.983622 + 1.844083\mathrm{i}$11

Characteristic polynomial

$$\begin{equation}\left(\lambda + 7\right)^{3} \left(\lambda^{6} - 52 \lambda^{5} - 592 \lambda^{4} + 11335 \lambda^{3} + 206414 \lambda^{2} + 1134572 \lambda + 2145391\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 7\right)^{3} \left(\lambda^{6} - 52 \lambda^{5} - 592 \lambda^{4} + 11335 \lambda^{3} + 206414 \lambda^{2} + 1134572 \lambda + 2145391\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 24 & 180 & 954 & 1911 & 2886 & 12192 & 25600 & 149200 \\ 1 & -1 & 9 & 168 & 136 & 208 & 1776 & 2496 & 19504 \\ 1 & 2 & 10 & 64 & 172 & 260 & 924 & 2238 & 12192 \\ 0 & 1 & 1 & 9 & 7 & 11 & 160 & 184 & 1728 \\ 0 & 0 & 1 & 2 & -2 & 6 & 16 & 56 & 288 \\ 0 & 0 & 1 & 2 & 8 & 6 & 32 & 96 & 444 \\ 0 & 0 & 0 & \frac{5}{2} & 1 & \frac{3}{2} & 9 & 14 & 168 \\ 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 2 & 0 & 24 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & \frac{3}{2} & 0\end{pmatrix}\notag\end{equation}$$