D.5.3.5 control_Matrix

Procedure from library equising.lib (see equising_lib).

Assume:

L is the output of multsequence(reddevelop(f)).

Return:

list M of 4 intmat’s:

M[1] contains the multiplicities at the respective infinitely near points p[i,j] (i=step of blowup+1, j=branch) – if branches j=k,...,k+m pass through the same p[i,j] then the multiplicity is stored in M[1][k,j], while M[1][k+1]=...=M[1][k+m]=0. M[2] contains the number of branches meeting at p[i,j] (again, the information is stored according to the above rule) M[3] contains the information about the splitting of M[1][i,j] with respect to different tangents of branches at p[i,j] (information is stored only for minimal j>=k corresponding to a new tangent direction). The entries are the sum of multiplicities of all branches with the respective tangent. M[4] contains the maximal sum of higher multiplicities for a branch passing through p[i,j] ( = degree Bound for blowing up)

Note:

the branches are ordered in such a way that only consecutive branches can meet at an infinitely near point.
the final rows of the matrices M[1],...,M[3] is (1,1,1,...,1), and correspond to infinitely near points such that the strict transforms of the branches are smooth and intersect the exceptional divisor transversally.

See also: multsequence.