A computational example
This Jupyter Notebook accompanies the paper “Mukai lifting of self-dual points” by Barbara Betti and Leonie Kayser. The code is in Julia language. The folder \(\texttt{MukaiLiftSource}\) contains all the files and we need to work inside this directory. We describe how to use the package \(\texttt{MukaiLiftP6}\).
[1]:
using Pkg;
Pkg.activate("MukaiLiftP6")
Activating project at `~/Documents/GitHub/Gr26/MukaiLiftSource/MukaiLiftP6`
[2]:
using MukaiLiftP6
using Oscar
using LinearAlgebra
using HomotopyContinuation
using DelimitedFiles
using JLD
Module MukaiLiftP6 done!
We define the configuration of points as a \(7\times 14\) matrix \(\Gamma\). We prove that it is self-dual by computing the matrix \(\Lambda\) satisfying condition (ii) of Lemma 2.1 and checking that \(\Gamma\cdot\Lambda\cdot\Gamma^\intercal=0\).
[9]:
Gamma = [7 -2 6 -1 -6 1 -9 7 0 6 1 8 -3 7; -1 2 -5 -2 0 -4 3 -3 -4 -3 4 -2 4 -1; 1 4 -1 -5 -3 6 8 -1 -8 -3 5 1 -6 -8; 3 -6 4 -3 -4 6 0 5 8 2 3 2 -8 0; 1 -2 1 0 -4 2 2 3 4 -1 2 2 -2 -2; 0 -6 -5 6 3 7 -3 2 8 -7 -6 -3 -5 5; -3 3 -4 1 4 3 2 -3 -6 -4 -3 -4 -1 -2]
Lambda = certify_selfdual(Gamma)
norm( Gamma*Lambda*transpose(Gamma), Inf)
1.7763568394002505e-14
We compute the Orthogonal normal form \(\Gamma_{\text{ONF}}= [I_7 \mid P]\) of \(\Gamma\) together with the linear projective transformation \(A\) such that \(A\cdot \Gamma_{\text{ONF}}= \Gamma\cdot \Lambda_{\text{scale}}\).
[10]:
Gamma_ONF, A, Lambda_scale = orthogonal_normal_form(Gamma)
norm(A*Gamma_ONF - Gamma*Lambda_scale, Inf)
2.220446049250313e-15
The orthogonal normal form of \(\Gamma\) is given by \([I_7 \mid P]\), where \(P\) is the orthogonal matrix:
[11]:
P = Matrix{Float64}(Gamma_ONF[:,8:14])
7×7 Matrix{Float64}:
-0.339885 0.059965 -0.0924461 … 0.756392 -0.303323 0.391105
-0.205214 -0.850587 0.0196552 0.26801 0.19422 -0.292163
0.100775 0.230199 0.811776 0.314816 -0.13109 -0.320813
0.094778 0.318981 -0.408428 0.329141 0.303156 -0.0306488
-0.773382 0.0614641 0.155243 -0.374698 -0.160906 0.200883
0.26268 -0.166903 -0.242692 … -0.071958 -0.855751 -0.155243
-0.395103 0.294162 -0.286833 0.0549679 -0.0546348 -0.770301
We compute the Telen norma form \(\Gamma_{\text{SNF}}\) of \(\Gamma\) using the Cayley transform of \(P\).
[12]:
S = cayley(P)
Gamma_SNF = [I+S I-S]
7×14 Matrix{Float64}:
1.0 -1.0 1.0 -1.0623e-14 … 1.0623e-14 1.0 -4.0 -2.0
1.0 1.0 3.0 -2.0 2.0 2.0 -10.0 -12.0
-1.0 -3.0 1.0 2.0 -2.0 -1.0 1.0 2.0
-6.26471e-15 2.0 -2.0 1.0 1.0 1.0 10.0 6.0
1.0 2.0 -1.0 1.0 -1.0 1.0 4.0 4.0
-4.0 -10.0 1.0 10.0 … -10.0 -4.0 1.0 6.0
-2.0 -12.0 2.0 6.0 -6.0 -4.0 -6.0 1.0
The target parameters \(S_{\text{target}}\) of the homotopy are the entries of the skew-symmetric matrix \(S\).
[13]:
S_target = skew_to_vector(S)
21-element Vector{Any}:
-0.9999999999999816
0.9999999999999984
-1.0623014660713928e-14
-1.000000000000007
3.999999999999994
2.000000000000003
2.999999999999989
-2.000000000000049
-2.00000000000003
9.99999999999996
⋮
1.0000000000000049
-0.9999999999999925
-2.0000000000000013
-0.9999999999999869
-9.999999999999995
-6.000000000000012
-3.9999999999999893
-3.9999999999999964
-5.999999999999995
[3]:
S_start, L_start = make_start();
parametrized_system, l_start, A_rand = make_poly_system(S_start, L_start);
We solve the mukai lifting problem for \(\Gamma\).
[14]:
@time result = HomotopyContinuation.solve(parametrized_system, l_start; compile = false, start_parameters=S_start, target_parameters=S_target)
6650.590251 seconds (99.43 M allocations: 39.405 GiB, 0.09% gc time, 0.63% compilation time)
Result with 1 solution
======================
• 1 path tracked
• 1 non-singular solution (0 real)
• random_seed: 0x559abd3b
We analyze the solution. There is one path tracked. The residual is the infinity norm of the vector with the equations evaluated in the computed solution and it is numerically zero.
[15]:
sol = solutions(result)[1]
result.path_results
1-element Vector{PathResult}:
PathResult:
• return_code → :success
• solution → ComplexF64[-0.25682992026101414 - 0.35352382958954476im, 0.07969833194871812 - 0.11571018739124789im, -0.09803109836006561 - 0.10914260579734429im, 0.10845446432945789 + 0.1213834741397078im, 0.07054820840239133 + 0.12372182781912008im, 0.034322032342818026 + 0.1287167678979342im, 0.10725708940751046 + 0.4343412410485293im, -0.29792718455039213 - 0.13280424269497115im, -0.07658176136159506 - 0.27576406865601im, -0.09985492528086015 + 0.02190214225580802im … -0.15447266899745862 - 0.09892023294732782im, -0.008805556396581355 + 0.1911048519825844im, 0.0030346242894781082 - 0.17269615245001688im, -0.23137949583152712 - 0.2546612734594592im, -0.010953640238052193 - 0.0934920698599431im, -0.02222878869905518 + 0.19759109538418715im, -0.1405859227250611 - 0.0526613568904753im, 0.16367653788044578 - 0.29963375910458334im, -0.3184718168060914 - 0.1617058275439371im, -0.19788431876656168 - 0.01693724052181073im]
• accuracy → 9.2136e-17
• residual → 7.5839e-13
• condition_jacobian → 2581.1
• steps → 1429 / 0
• extended_precision → false
• path_number → 1
The embedding \(\widehat L\colon {\mathbb P}^6 \simeq \mathbb{L} \subseteq {\mathbb P}^{14}\) such that \(\widehat L(\Gamma_{\text{SNF}})=\mathbb{L}\cap X_8\) is:
[16]:
L_hat = L_start + sum(sol[i]*A_rand[i] for i=1:69)
15×7 Matrix{ComplexF64}:
-2.39867-0.760585im 0.136927-0.0731426im … -0.0549966+1.54306im
0.120381-0.687571im 0.118839+0.0175647im 0.609849-0.349064im
-0.874564+0.489957im -0.133184+0.161753im -0.484225-0.170822im
0.0900887-0.80893im 0.213471+0.539043im -0.336838+0.803372im
-0.660979-1.1459im 0.0312071+0.382946im -0.554239+0.55146im
1.91901+0.50087im 0.0779703+0.333896im … -1.3462+0.21279im
-0.266978-0.832334im -0.255807-0.494672im -2.21711+0.997617im
-0.22492-3.90594im -0.635566-0.176003im -1.23474-1.53634im
-0.408724-0.897665im -0.341599-0.401544im -1.70067-0.732123im
1.33645+1.10583im -0.00976897+0.302164im 0.557198+0.715766im
-0.113212-0.0633213im 0.419823-0.176421im … 0.508407+0.838852im
-0.0102615+0.157629im 0.208108+0.00825777im 0.733529+0.933496im
-0.187805-0.131456im 0.350144+0.326324im 1.62697+0.668133im
-0.559089-0.345816im 0.22217+0.313894im 1.03892+0.242524im
-0.789949+0.774166im 0.00451075-0.041194im -0.418487-0.0969304im
Composing with the previous coordinate transformations \(\Gamma \leadsto \Gamma_{\text{SNF}}\) we obtain \(L \colon \mathbb{P}^6 \to \mathbb{P}^{14}\) such that \(L(\Gamma)= \mathbb{L}\cap X_8\)
[17]:
L = L_hat*(I+S)*inv(A)
15×7 Matrix{ComplexF64}:
-2.43381-4.45071im -2.86825-4.49986im … 1.85726-4.12053im
-0.571879+1.44484im -0.213834+1.77774im -2.17004+0.819291im
0.495441-0.523548im 1.01963-0.898375im 2.63775+0.589267im
0.958401-2.80583im 0.350694-3.20325im -0.734016-3.48259im
0.505669-2.34915im 0.0221898-2.41816im 1.21409-1.60625im
2.02096-3.16752im 1.04499-4.93784im … 2.92458-2.06121im
5.18851+1.67306im 4.03309+4.42388im 5.8118-0.786245im
3.18389+1.1179im 4.84222+1.73233im 8.57393+9.09686im
3.93173+1.05756im 4.70829+0.929709im 8.69331+3.16074im
-1.17617-1.68854im -1.02489-2.27274im -1.17072-3.32841im
-2.54915-0.570031im -3.52059+0.245008im … -2.94125-1.78315im
-2.25506-0.596842im -2.57807+0.322309im -2.8166-2.26911im
-1.30788-2.48251im -0.16074-2.09195im -6.09222-1.63651im
-1.31124-1.72178im -0.557748-1.70476im -3.54458-0.648492im
0.239302+0.778868im 0.255953+0.461925im 2.3461-0.475216im
We can verify \(L(\Gamma) = \mathbb{L} \cap X_8\) by evaluating \(L(\gamma_i)\) in the Plücker relations.
[18]:
plück_oscar = gens( grassmann_pluecker_ideal(2,6))
q = poly_to_func(plück_oscar)
maximum([ norm(q(L*Gamma[:,i]) ,Inf) for i =1:14])
1.2805801671064476e-11