Singular Manual: blowUpBO

| #### D.5.15.8 blowUpBO Procedure from library `resolve.lib` (see resolve_lib). Usage: blowUpBO (BO,C,e); BO = basic object, a list: ideal W, ideal J, intvec b, list Ex, ideal ab, intvec v, intvec w, matrix M C = ideal e = integer (0 usual blowing up, 1 deleting extra charts, 2 deleting no charts ) Assume: R = basering, a polynomial ring, W an ideal of R, J = ideal containing W, C = ideal containing J Compute: the blowing up of BO[1] in C, the exceptional locus, the strict transform of BO[2] Note: blowUpBO may be applied to basic objects in the sense of [Bravo, Encinas, Villamayor] in the following referred to as BO and to presentations in the sense of [Bierstone, Milman] in the following referred to as BM. Return: a list l of length at most size(C), l[i] is a ring containing an object BO resp. BM: BO[1]=BM[1] an ideal, say Wi, defining the ambient space of the i-th chart of the blowing up BO[2]=BM[2] an ideal defining the strict transform BO[3] intvec, the first integer b such that in the original object (Delta^b(BO[2]))==1 the subsequent integers have the same property for Coeff-Objects of BO[2] used when determining the center BM[3] intvec, BM[3][i] is the assigned multiplicity of BM[2][i] BO[4]=BM[4] the list of exceptional divisors BO[5]=BM[5] an ideal defining the map K[W] ----> K[Wi] BO[6]=BM[6] an intvec BO[6][j]=1 indicates that <BO[4][j],BO[2]>=1, i.e. the strict transform does not meet the j-th exceptional divisor BO[7] intvec, the index of the first blown-up object in the resolution process leading to this object for which the value of b was BO[3] the subsequent ones are the indices for the Coeff-Objects of BO[2] used when determining the center BM[7] intvec, BM[7][i] is the index at which the (2i-1)st entry of the invariant first reached its current maximal value BO[8]=BM[8] a matrix indicating that BO[4][i] meets BO[4][j] by BO[8][i,j]=1 for i < j BO[9] empty BM[9] the invariant Example:
    LIB "resolve.lib";
    ring R=0,(x,y),dp;
    ideal W;
    ideal J=x2-y3;
    intvec b=1;
    list E;
    ideal abb=maxideal(1);
    intvec v;
    intvec w=-1;
    matrix M;
    intvec ma;
    list BO=W,J,b,E,abb,v,w,M,ma;
    ideal C=CenterBO(BO)[1];
    list blow=blowUpBO(BO,C,0);
    def Q=blow[1];
    setring Q;
    BO;
    ==> [1]:
    ==>    _[1]=0
    ==> [2]:
    ==>    _[1]=y(1)^2-x(2)
    ==> [3]:
    ==>    1
    ==> [4]:
    ==>    [1]:
    ==>       _[1]=x(2)
    ==> [5]:
    ==>    _[1]=x(2)*y(1)
    ==>    _[2]=x(2)
    ==> [6]:
    ==>    0
    ==> [7]:
    ==>    -1
    ==> [8]:
    ==>    _[1,1]=0
    ==> [9]:
    ==>    0

User manual for Singular version 4.4.0, 2024, generated by texi2html.