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D.4.19.6 primeClosure
Procedure from library normal.lib (see normal_lib).
- Usage:
- primeClosure(L [,c]); L a list of a ring containing a prime ideal
ker, c an optional integer
- Return:
- a list L (of size n+1) consisting of rings L[1],...,L[n] such that
- L[1] is a copy of (not a reference to!) the input ring L[1]
- all rings L[i] contain ideals ker, L[2],...,L[n] contain ideals phi
such that
L[1]/ker --> ... --> L[n]/ker
are injections given by the corresponding ideals phi, and L[n]/ker
is the integral closure of L[1]/ker in its quotient field.
- all rings L[i] contain a polynomial nzd such that elements of
L[i]/ker are quotients of elements of L[i-1]/ker with denominator
nzd via the injection phi.
L[n+1] is the delta invariant
- Note:
- - L is constructed by recursive calls of primeClosure itself.
- c determines the choice of nzd:
- c not given or equal to 0: first generator of the ideal SL,
the singular locus of Spec(L[i]/ker)
- c<>0: the generator of SL with least number of monomials.
Example:
| | LIB "normal.lib";
ring R=0,(x,y),dp;
ideal I=x4,y4;
def K=ReesAlgebra(I)[1]; // K contains ker such that K/ker=R[It]
list L=primeClosure(K);
def R(1)=L[1]; // L[4] contains ker, L[4]/ker is the
def R(4)=L[4]; // integral closure of L[1]/ker
setring R(1);
R(1);
==> // coefficients: QQ
==> // number of vars : 4
==> // block 1 : ordering dp
==> // : names x y U(1) U(2)
==> // block 2 : ordering C
ker;
==> ker[1]=y^4*U(1)-x^4*U(2)
setring R(4);
R(4);
==> // coefficients: QQ
==> // number of vars : 10
==> // block 1 : ordering a
==> // : names X(1) X(2) X(3) X(4) X(5) X(6) X(7) T(2) T(\
3) T(4)
==> // : weights 1 1 1 1 3 2 2 1 \
1 1
==> // block 2 : ordering dp
==> // : names X(1) X(2) X(3) X(4) X(5) X(6) X(7) T(2) T(\
3) T(4)
==> // block 3 : ordering C
ker;
==> ker[1]=X(2)*X(7)-X(5)
==> ker[2]=X(1)*X(6)-X(5)
==> ker[3]=X(1)^2*X(4)-X(2)*X(6)
==> ker[4]=X(2)^2*X(3)-X(1)*X(7)
==> ker[5]=X(1)*X(4)*X(7)-X(6)^2
==> ker[6]=X(1)*X(2)*X(3)*X(4)-X(6)*X(7)
==> ker[7]=X(2)*X(3)*X(6)-X(7)^2
==> ker[8]=X(1)*T(3)-X(2)*T(4)
==> ker[9]=-X(2)*T(2)+X(1)*T(4)
==> ker[10]=-X(2)*T(2)+X(7)
==> ker[11]=-X(1)*T(3)+X(6)
==> ker[12]=X(6)*T(2)-X(7)*T(4)
==> ker[13]=X(2)*X(4)*T(2)-X(6)*T(3)
==> ker[14]=X(1)*X(4)-X(2)*T(3)
==> ker[15]=X(1)*X(4)*T(2)-X(6)*T(4)
==> ker[16]=X(2)*X(3)-X(1)*T(2)
==> ker[17]=-X(1)^2*T(3)+X(5)
==> ker[18]=X(3)*X(6)-X(7)*T(2)
==> ker[19]=X(2)*X(3)*X(4)-X(7)*T(3)
==> ker[20]=X(1)*X(3)*X(4)-X(7)*T(4)
==> ker[21]=X(2)^2*X(6)*T(4)-X(2)*X(5)*T(3)
==> ker[22]=T(2)^2-X(3)*T(4)
==> ker[23]=-X(3)*X(4)+T(2)*T(3)
==> ker[24]=T(3)^2-X(4)*T(4)
==> ker[25]=-X(3)*T(3)+T(2)*T(4)
==> ker[26]=-X(4)*T(2)+T(3)*T(4)
==> ker[27]=-X(3)*X(4)+T(4)^2
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