Fano fourfolds of type A beta

zero loci on products of type A Grassmannians and flag varieties

Family 4-192-45-a

·

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

Defining data
Ambient
ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ2P2
node 1: fundamental weight ω_1 (marked)node 2: fundamental weight ω_2 (marked)node 3: fundamental weight ω_3 (marked)node 4: fundamental weight ω_4 (marked)node 5: fundamental weight ω_5ω1ω2ω3ω4ω5
Bundle
E^(w1 + w2 + w3 + w4)
Bundle
O_1(1) box O_2(1) box O_3(1) box O_4(1)
ZL62 label
11120.x5000
Line bundles on the ambient $Y$
$$\begin{equation} \notag \begin{aligned} \omega_Y^\vee &\cong \mathcal{O}(2;2;2;3) \\ \det\mathcal{E} &\cong \mathcal{O}(1;1;1;1) \end{aligned} \end{equation}$$
Numerical invariants
Fano index = 1
$\mathrm{h}^0(X, \omega_X^\vee)$ = 45
$\operatorname{vol}(X) = \mathrm{c}_1^4$ = 192
$\chi(X, \mathrm{T}_X)$ = -6

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

Chern numbers of $\mathrm{T}_X$
$\mathrm{c}_1^4$ = 192
$\mathrm{c}_1^2\mathrm{c}_2$ = 144
$\mathrm{c}_2^2$ = 102
$\mathrm{c}_1\mathrm{c}_3$ = 52
$\mathrm{c}_4$ = 22
Hodge diamond
1
00
0 4 0
0 0 0 0
0 0 12 0 0
0 0 0 0
0 4 0
00
1
Polyvector parallelogram & Hochschild cohomology
$\mathrm{HH}^{ 0 }$ = 1 1
$\mathrm{HH}^{ 1 }$ = 0 0 0
$\mathrm{HH}^{ 2 }$ = x0 + 6 0 6 x0
$\mathrm{HH}^{ 3 }$ = x0 + 30 0 0 x0 30
$\mathrm{HH}^{ 4 }$ = 45 0 0 0 0 45
$\mathrm{HH}^{ 5 }$ = 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

$\operatorname{Hilb}_X(t)$ $=$
$\displaystyle\frac{ 1 + 40t + 110t^{2} + 40t^{3} + t^{4} }{(1-t)^{ 5 }}$
$=$

Regularized quantum period

Quantum period in the literature

cataloguematching entries
Kalashnikov (2019) CKK(405)
Coates et al. (2020) No match
Coates–Kasprzyk–Prince (2015) CKP(475)

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( 3t + 1 \right) \left( 91t^{4} + 24t^{3} - 22t^{2} - 9t - 1 \right) \left( 1163t^{4} + 120t^{3} - 46t^{2} + 15t - 1 \right) \left( 134810100590055t^{13} + 210268376810652t^{12} + 120791215754842t^{11} + 126715573834756t^{10} + 103020888950204t^{9} + 41318309663588t^{8} + 8007383234412t^{7} + 745785740009t^{6} + 46257195912t^{5} + 5907172509t^{4} + 458916046t^{3} + 7645278t^{2} + 1175t + 544 \right)D^{6} + \left( 898843514672079321345t^{22} + 1978247298637582133271t^{21} + 1650219797876436018318t^{20} + 1273257842026501159316t^{19} + 1157178090881862369760t^{18} + 630301476585793874070t^{17} + 84288351127251910357t^{16} - 87344627076946332663t^{15} - 54831355271475468239t^{14} - 15202429487521415805t^{13} - 2435026179747775646t^{12} - 252688180168968412t^{11} - 22227825720698191t^{10} - 2842122361384613t^{9} - 539195118921740t^{8} - 83991031943788t^{7} - 7498498302088t^{6} - 421515760318t^{5} - 36677138983t^{4} - 2296499445t^{3} - 30862077t^{2} - 6789t - 1088 \right)D^{5} + \left( 7490362622267327677875t^{22} + 16131202288330916757045t^{21} + 13485265035683515400586t^{20} + 11824171661596270907132t^{19} + 11273971708214186711036t^{18} + 6386861710797810404806t^{17} + 1477136174427374983725t^{16} - 242413073194519957810t^{15} - 246922553624347890480t^{14} - 69292798092925419736t^{13} - 10412348851049254185t^{12} - 993351954940794778t^{11} - 80414742944746089t^{10} - 5083309541900212t^{9} + 396923199311911t^{8} + 130913230516568t^{7} + 11301954439358t^{6} + 515583760273t^{5} + 42132644096t^{4} + 2110880679t^{3} + 24882991t^{2} - 5147t + 544 \right)D^{4} + t \left( 31459523013522776247075t^{21} + 66582216449948367012885t^{20} + 55398962503293931351290t^{19} + 52928766802656556145900t^{18} + 51756947582259025259272t^{17} + 29827931464634706699122t^{16} + 8235312618817139145939t^{15} + 227169736791889265523t^{14} - 520130883880179530457t^{13} - 158900589239324210837t^{12} - 23612783010707693824t^{11} - 2479421149994028328t^{10} - 293339684816779229t^{9} - 34693599967082915t^{8} - 2568861033048456t^{7} - 81788679131554t^{6} + 1860107019082t^{5} + 222835962544t^{4} + 3127506357t^{3} + 35123821t^{2} - 985535t - 663 \right)D^{3} + 2 t ^{2} \left( 34755282567320400425340t^{20} + 72569553953824561187616t^{19} + 59932407894653811400332t^{18} + 60835768845989924448508t^{17} + 60400289726066881830526t^{16} + 35048627817519434601646t^{15} + 10438981726215333280197t^{14} + 1051815857020039108593t^{13} - 251645435396826565381t^{12} - 96623448448178467517t^{11} - 14733304273344843140t^{10} - 1790778298254024944t^{9} - 262476078989876361t^{8} - 31103701491899485t^{7} - 1966841134803644t^{6} - 34792265648488t^{5} + 2874119985393t^{4} + 198772718651t^{3} + 1024144604t^{2} - 5918662t - 405376 \right)D^{2} + 4 t ^{2} \left( 18875713808113665748245t^{20} + 39019576219140567529671t^{19} + 31982444866866361180098t^{18} + 33880333542278989385336t^{17} + 33964228833720674155777t^{16} + 19744714614165937814737t^{15} + 6087768693163168732091t^{14} + 816127435923087981989t^{13} - 39129087633056826212t^{12} - 29298099171222196482t^{11} - 4788571708349970359t^{10} - 697803766974708055t^{9} - 118530978132535477t^{8} - 13750806568924057t^{7} - 773928908264011t^{6} - 7281305028125t^{5} + 1293357801613t^{4} + 74378030967t^{3} + 428237276t^{2} - 3329956t - 91392 \right)D + 8 t ^{2} \left( 3852186491451768520050t^{20} + 7908965779368136074390t^{19} + 6443677761166860265188t^{18} + 7022272637109905975136t^{17} + 7082067668958621863475t^{16} + 4116702318723190331839t^{15} + 1289829912233640249334t^{14} + 193981192561843646210t^{13} + 2937822739258323875t^{12} - 3417055358677472931t^{11} - 628454906585759081t^{10} - 111264860833311229t^{9} - 20913802655349411t^{8} - 2375584543605689t^{7} - 124054529783461t^{6} - 692994533343t^{5} + 202564702373t^{4} + 10197168177t^{3} + 75458498t^{2} - 326808t - 8704 \right) \notag \end{equation}$$

Quantum spectrum

Ambient: ℙ1P1 × ℙ1P1 × ℙ1P1 × ℙ2P2

|λ| = 12.68936 λ ≈ 12.68936; multiplicity 1; Jordan blocks 1 λ ≈ 2.600426 - 4.994878i; multiplicity 1; Jordan blocks 1 λ ≈ 2.600426 + 4.994878i; multiplicity 1; Jordan blocks 1 λ ≈ -4.234552 - 1.940705i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -4.234552 + 1.940705i; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ -3; multiplicity 1; Jordan blocks 1 λ ≈ -2.890212; multiplicity 1; Jordan blocks 1 λ ≈ -2.3305; multiplicity 3; Jordan blocks 1, 1, 1 λ ≈ 1.799605; multiplicity 3; Jordan blocks 1, 1, 1 3 3 3 3
Sector dimension
17
Spectral radius
$12.6893597$
Diagonalizable at 1
Yes
Simple spectrum at 1
No
Eigenvalue approximations and Jordan block sizes
eigenvaluemultiplicityJordan blocks
$12.68936$11
$2.600426 - 4.994878\mathrm{i}$11
$2.600426 + 4.994878\mathrm{i}$11
$-4.234552 - 1.940705\mathrm{i}$31, 1, 1
$-4.234552 + 1.940705\mathrm{i}$31, 1, 1
$-3$11
$-2.890212$11
$-2.3305$31, 1, 1
$1.799605$31, 1, 1

Characteristic polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 9 \lambda^{3} + 22 \lambda^{2} - 24 \lambda - 91\right)^{3} \left(\lambda^{4} - 15 \lambda^{3} + 46 \lambda^{2} - 120 \lambda - 1163\right)\notag\end{equation}$$

Minimal polynomial

$$\begin{equation}\left(\lambda + 3\right) \left(\lambda^{4} + 9 \lambda^{3} + 22 \lambda^{2} - 24 \lambda - 91\right) \left(\lambda^{4} - 15 \lambda^{3} + 46 \lambda^{2} - 120 \lambda - 1163\right)\notag\end{equation}$$

Quantum multiplication matrix

$$\begin{equation}\begin{pmatrix}0 & 4 & 4 & 4 & 2 & 6 & 6 & 3 & 6 & 3 & 3 & 9 & 0 & 8 & 8 & 8 & 30 \\ 1 & -1 & 1 & 1 & 0 & 2 & 2 & 2 & 2 & 0 & 0 & 2 & 0 & 3 & 3 & 0 & 8 \\ 1 & 1 & -1 & 1 & 0 & 2 & 2 & 0 & 2 & 2 & 0 & 2 & 0 & 3 & 0 & 3 & 8 \\ 1 & 1 & 1 & -1 & 0 & 2 & 2 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 3 & 3 & 8 \\ 2 & 1 & 1 & 1 & 0 & 2 & 2 & 4 & 2 & 4 & 4 & 0 & 6 & 6 & 6 & 6 & 0 \\ 0 & 1 & 1 & 0 & 0 & -2 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 & 0 & 1 & -2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 \\ 0 & 2 & 0 & 0 & 1 & 1 & 1 & -1 & 0 & 1 & 1 & 0 & 2 & 2 & 2 & 2 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 \\ 0 & 0 & 2 & 0 & 1 & 1 & 0 & 1 & 1 & -1 & 1 & 0 & 2 & 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 & 1 & 1 & 1 & 1 & -1 & 0 & 2 & 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 2 & 2 & 2 & 6 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & -4 & 1 & -4 & -4 & -6 & -3 & -4 & -4 & -4 & -10 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 3 & 0 & 3 & 0 & 2 & 1 & 0 & 2 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 0 & 0 & 3 & 2 & 1 & 2 & 0 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 2 & 1 & 2 & 2 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 3 & 3 & 3 & 0\end{pmatrix}\notag\end{equation}$$