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### Integral closure of an ideal

We want to compute the integral closure of the Jacobian ideal of the -singularity defined by . Again, we set the printlevel sufficiently high to get intermediate results and comments, which are partly reproduced.


LIB "reesclos.lib";
ring A = 0,(x,y,z),dp;
ideal I = jacob(x5+y3+z2);  // the Jacobian ideal

Let us first compute the Rees algebra of .

list rees = ReesAlgebra(I);
def Rees = rees[1];
setring Rees;
reesid;
==> reesid[1]=3*y^2*U(3)-2*z*U(2)
==> reesid[2]=5*x^4*U(3)-2*z*U(1)
==> reesid[3]=5*x^4*U(2)-3*y^2*U(1)
def At = rees[2]; setring At;
reesmap;
==> reesmap[1]=x     reesmap[2]=y     reesmap[3]=z
==> reesmap[4]=5x4t  reesmap[5]=3y2t  reesmap[6]=2zt

reesid is isomorphic to the Rees algebra as subalgebra of , under the map

reesmapreesmap

Let us now compute the integral closure of I:

list norI = normalI(I);

After 3 iterations we reach the normalization of the Rees algebra as modulo the ideal

==> 4*T(4)*T(5)-15*T(7)^2
==> 5*T(1)^2*T(7)-2*T(2)*T(4)
==> 3*T(2)^2*T(6)-2*T(3)*T(5)
==> 2*T(1)^2*T(5)-3*T(2)*T(7)
==> 4*T(1)*T(4)*T(5)*T(7)-15*T(1)*T(7)^3
==> T(1)^2*T(2)*T(6)-T(3)*T(7)
==> 5*T(1)^4*T(6)-2*T(3)*T(4)

Now we have to determine the map . This is computed by representing the ring variables as fractions in the variables of the Rees algebra. We get

==> T(1) : 25*x^9*z
==> T(2) : 25*x^8*y*z
==> T(2) : 25*x^8*z^2
==> T(2) : 25*x^8*z*U(1)
==> T(2) : 15*x^4*y^2*z*U(1)
==> T(2) : 10*x^4*z^2*U(1)
==> T(2) : 10*x^6*y*z*U(1)

with the universal'' denominator: . Since is the image under the map , , , , is generated in by

==> generator 1 : x       generator 2 : y
==> generator 3 : z       generator 4 : 5x4t
==> generator 5 : 3y2t    generator 6 : 2zt
==> generator 7 : 2x2yt

That is, is generated in -degree 1 and, hence, for all . In particular, the integral closure of is generated by 4 elements, the extra element being . This result is stored in the first entry of the list norI:

norI[1];
==> _[1]=5x4  _[2]=3y2  _[3]=2z  _[4]=2x2y


Next: Effective Construction of Algebraic Up: Examples Previous: Non-normal locus
Christoph Lossen
2001-03-21