2. Normalisation

Another important algorithm is the normalisation of $K[x]/I$ where $I$ is a radical ideal. It can be used as a step in the primary decomposition, as proposed in EHV, but is also of independent interest. Several algorithms have been proposed, especially by Se, Sto, GT, Va1. It had escaped the computer algebra community, however, that GR had given a constructive proof for the ideal of the non-normal locus of a complex space. Within this proof they provide a normality criterion which is essentially an algorithm for computing the normalisation, cf. Jo. Again, to make the algorithm efficient needed some extra work which is described in DGJP. The Grauert-Remmert algorithm is implemented in SINGULAR and seems to be the only full implementation of the normalisation.

Criterion [Grauert and Remmert1971]: Let $R = K[x]/I$ with $I$ a radical ideal. Let $J$ be a radical ideal containing a non-zero divisor of $R$ such that $V(J)$ contains the non-normal locus of $V(I)$. Then $R$ is normal if and only if $R =$   Hom$_R(J,J)$.

For $J$ we may take any ideal so that $V(J)$ contains the singularities of $V(I)$. Since normalisation commutes with localisation, we obtain

Corollary: Ann $($Hom$_R(J,J)/R)$ is an ideal describing the non-normal locus of $V(I)$.

Now Hom$_R(J,J)$ is a ring containing $R$ and if $R \subsetneqq$   Hom$_R(J,J)=R_1$ we can continue with $R_1$ instead of $R$ and obtain an increasing sequence of rings $R \subset R_1 \subset R_2 \subset \ldots$.

After finitely many steps the sequence becomes stationary (because the normalisation of $R = K[x]/I$ is finite over $R$) and we reach the normalisation of $R$ by the criterion of Grauert and Remmert.

Ingredients for the normalisation (which is a highly recursive algorithm):

1. Computation of the ideal $J$ of the singular locus of the ideal $I$;
2. computation of a non-zero divisor for $J$;
3. ring structure on Hom$(J,J)$;
4. syzygies, normal forms, ideal quotient.

SINGULAR commands for computation of the normalisation:

LIB "normal.lib";
ring S   = 0,(x,y,z),dp;
ideal I  = y2-x2z;
list nor = normal(I);
def R    = nor[1];
setring R;
normap;
==> normap[1]=T(1)
==> normap[2]=T(1)*T(2)
==> normap[3]=T(2)^2

$\parbox{8.5cm}{ \unitlength0.75cm \begin{picture}(8.5,4) \put(-1.... ...\put(6.5,-0.3){\includegraphics[width=3.5cm]{Bilder/normal4.ps}} \end{picture}}$

In the preceding picture, $R$, the normalisation of $S$, is just the polynomial ring in two variables $T(1)$ and $T(2)$. (The ``handle'' of Whitney's umbrella is invisible in the parametric picture since it requires an imaginary parameter $t$.)

In several cases the normalisation of a variety is smooth (for example, the normalisation of the discriminant of a versal deformation of an isolated hypersurface singularity) sometimes even an affine space. In this case, the normalisation map provides a parametrisation of the variety. This is the case for the Whitney umbrella: $V = \{y^2 - zx^2 = 0\}$.