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kGonalNodalCurves :: lineBundleFromPointsAndMultipliers

lineBundleFromPointsAndMultipliers -- computes basis of a line bundle from the 2g points P_i, Q_i and the multipliers

Synopsis

Description

If C is a g-nodal canonical curve with normalization ν: PP1 →PPg-1 then a line bundle L of degree k on C is given by ν*(OOPP1(k))≅L and gluing data (bj)/(aj):OOPP1⊗kk(Pj)→OOPP1⊗kk(Qj). Given 2g points Pi, Qi and the multipliers (ai,bi) we can compute a basis of sections of L as a kernel of the matrix A=(A)ij with Aij=biBj(Pi)-aiBj(Qi) where Bj:PP1→kk, (p0:p1)→p0k-jp1j.

i1 : (k,g,n)=(4,8,1000);
i2 : time (P,Q,multL,f)=pickGoodPoints(k,g,n);
     -- used 0.147619 seconds
i3 : time f'=lineBundleFromPointsAndMultipliers(multL,P,Q,k);
     -- used 0.026673 seconds

              ZZ          1        ZZ          2
o3 : Matrix (----[x , x ])  <--- (----[x , x ])
             1009  0   1          1009  0   1
i4 : ideal f==ideal f'

o4 = true

See also

Ways to use lineBundleFromPointsAndMultipliers :

  • lineBundleFromPointsAndMultipliers(List,Matrix,Matrix,ZZ)