2. Introduction by pictures

The basic problem of algebraic geometry is to understand the set of points $x = (x_1, \dots, x_n) \in K^n$ satisfying a system of equations

$$\begin{array}{ccc} f_1(x_1, \dots, x_n) & = & 0,\\ \vdots & & \\ f_k(x_1, \dots, x_n) & = & 0, \end{array}$$

where $K$ is a field and $f_1, \dots, f_k$ are elements of the polynomial ring $K[x] = K[x_1, \dots, x_n]$.

The solution set of $f_1 = 0, \dots, f_k= 0$ is called the algebraic set, or algebraic variety of $f_1, \dots, f_k$ and is denoted by

$$V = V(f_1, \dots, f_k).$$

It is easy to see, and important to know, that $V$ depends only on the ideal

$$I = \langle f_1, \dots, f_k\rangle = \{f \in K[x]\mid f = \sum^k_{i=1} a_i f_i,\; a_i \in K[x]\}$$

generated by $f_1, \dots, f_k$ in $K[x]$, that is $V = V(I) = \{x \in K^n\mid f(x) = 0\; \forall\; f \in I\}$.

The Cayley Cubic There is a unique cubic surface which has four ordinary double points, usually called the Cayley cubic after its discoverer, Arthur Cayley. It is a degeneration of the Clebsch cubic, has S4 as symmetry group, and the projective equation is z0z1 z2 +z0 z1 z3 +z0 z2 z3 +z1 z2 z3 =0

A Cubic with a D4-Singularity Degenerating the Cayley cubic we receive a D4-singularity. The affine equation is 5x(x2-5y2)+z2(1+z)+2xy+2yz = 0.

The Barth Sextic The equation for this sextic was found by Wolf Barth. It has 65 ordinary double points, the maximal possible number for a sextic. Its affine equation is (with c=(1+sqrt(5))/2) 16(2c+1)x2y2z2 - 4c4(x4y2+ y4z2+x2z4) +4c2(x2y4+ y2z4+x4z2) -(2c+1)(x2+y2+ z2-1)2=0.

An Ordinary Node An ordinary node is the most simple singularity. It has the local equation x2+y2-z2=0.

Whitney's Umbrella The Whitney umbrella is named after Hassler Whitney who studied it in connection with the stratification of analytic spaces. It has the local equation x2-y2z=0.

A 5-nodal plane curve of degree 11 with equation -32x2+2097152y11-1441792y9+360448y7 -39424y5+1760y3-22y+1=0, a deformation of A10:   y11-x2=0.

Space Curve This space curve is given parametrically by x=t4, y=t3, z=t2, or implicitly by x-z2=y2-z3=0.

Of course, if for some polynomial $f \in K[x]$, $f^d\vert _V = 0$, then $f\vert _V = 0$ and hence, $V = V(I)$ depends only on the radical of $I$,

$$\sqrt{I} = \{f \in K[x] \mid f^d \in I,$$    for some $$d\}.$$

The biggest ideal determined by $V$ is

$$I(V) = \{f \in K[x] \mid f(x) = 0\; \forall\; x \in V\},$$

and we have $I \subset \sqrt{I} \subset I (V)$ and $V\bigl(I(V)\bigr) = V(\sqrt{I}) = V(I) = V$.

The important Hilbert Nullstellensatz states that, for $K$ an algebraically closed field, we have for any variety $V \subset K^n$ and any ideal $J \subset K[x]$,

$$V = V(J) \Rightarrow I(V) = \sqrt{J}$$

(the converse implication being trivial). That is, we can recover the ideal $J$, up to radical, just from its zero set and, therefore, for fields like ${\mathbb{C}}$ (but, unfortunately, not for ${\mathbb{R}}$) geometry and algebra are ``almost equal''. But almost equal is not equal and we shall have occasion to see that the difference between $I$ and $\sqrt{I}$ has very visible geometric consequences.

Many of the problems in algebra, in particular, computer algebra, have a geometric origin. Therefore, I choose an introduction by means of some pictures of algebraic varieties, some of them being used to illustrate subsequent problems.

The above pictures were not only chosen to illustrate the beauty of algebraic geometric objects but also because these varieties have had some prominent influence on the development of algebraic geometry and singularity theory.

The Clebsch cubic itself has been the object of numerous investigations in global algebraic geometry, the Cayley and the D₄-cubic also, but, moreover, since the D₄-cubic deforms, via the Cayley cubic, to the Clebsch cubic, these first three pictures illustrate deformation theory, an important branch of (computational) algebraic geometry.

The ordinary node, also called A₁-singularity (shown as a surface singularity) is the most simple singularity in any dimension. The Barth sextic illustrates a basic but very difficult and still (in general) unsolved problem: to determine the maximum possible number of singularities on a projective variety of given degree. In Section 7.3 we report on recent progress on this question for plane curves.

Whitney's umbrella was, at the beginning of stratification theory, an important example for the two Whitney conditions. We use the umbrella in Section 4.2 to illustrate that the algebraic concept of normalisation may even lead to a parametrisation of a singular variety, an ultimate goal in many contexts, especially for graphical representations. In general, however, such a parametrisation is not possible, even not locally, if the variety has dimension bigger than one. For curve singularities, on the other hand, the normalisation is always a parametrisation. Indeed, computing the normalisation of the ideal given by the implicit equations for the space curve in the last picture, we obtain the given parametrisation. Conversely, the equations are derived from the parametrisation by eliminating t, where elimination of variables is perhaps the most important basic application of Gröbner bases.

Finally, the 5-nodal plane curve illustrates the global existence problem described in Section 7.2. Moreover, these kind of deformations with the maximal number of nodes play also a prominent role in the local theory of singularities. For instance, from this real picture we can read off the intersection form and, hence, the monodromy of the singularity A₁₀ by a beautiful theory of A'Campo and Gusein-Zade. We shall present a completely different, algebraic algorithm to compute the monodromy in Section 6.3.

For more than a hundred years, the connection between algebra and geometry has turned out to be very fruitful and both merged to one of the leading areas in mathematics: algebraic geometry. The relationship between both disciplines can be characterised by saying that algebra provides rigour while geometry provides intuition.

In this connection, I place computer algebra on top of rigour, but I should like to stress its limited value if it is used without intuition.