Integral closure of an ideal

We want to compute the integral closure of the Jacobian ideal $I$ of the $E_8$-singularity defined by $x^5\!+y^3\!+z^2=0$. 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 $I$.

    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

$\mathbf{Q}[x,y,z,U_1,U_2,U_3]/$reesid is isomorphic to the Rees algebra $\mathcal{R}(I)$ as subalgebra of $\mathbf{Q}[x,y,z,t]$, under the map

$$(x,y,z,U_1,U_2,U_3) \longmapsto \bigl($$reesmap$$(1),\dots,$$reesmap$$(6)\bigr)\,.$$

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 $Q[T_1,\dots ,T_7]$ 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 $Q[T_1,\dots,T_7] \to A[t]$. This is computed by representing the ring variables $T_1,\dots,T_7$ 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: $25x^8z$. Since $\mathcal{R}(I)$ is the image under the map $\mathbf{Q}[x,y,z,U_1,U_2,U_3] \longrightarrow \mathbf{Q}[x,y,z,t]$, $U_1\mapsto 5x^4t$, $U_2\mapsto 3y^2t$, $U_3\mapsto 2zt$, $\mathcal{R}(I)$ is generated in $Q[x,y,z,t]$ by

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

That is, $\mathcal{R}(I)$ is generated in $t$-degree 1 and, hence, $\overline{I^k} = \bigl(\overline{I}\bigr)^k$ for all $k > 1$. In particular, the integral closure of $I = \langle 5x^4,3x^2,2z\rangle$ is generated by 4 elements, the extra element being $x^2y$. This result is stored in the first entry of the list norI:

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