3. Some problems in algebraic geometry

In this section I shall formulate some of the basic questions and problems arising in algebraic geometry and provide ingredients for certain algorithms. I shall restrict myself to those algorithms where I am somehow familiar with their implementations and which have turned out to be useful in practical applications.

Let me first recall the most basic but also most important applications of Gröbner bases to algebraic constructions (called ``Gröbner basics'' by Sturmfels). Since these can be found in more or less any textbook dealing with Gröbner bases, I just mention them:

• Ideal (resp. module) membership problem
• Intersection with subrings (elimination of variables)
• Intersection of ideals (resp. submodules)
• Zariski closure of the image of a map
• Solvability of polynomial equations
• Solving polynomial equations
• Radical membership
• Quotient of ideals
• Saturation of ideals
• Kernel of a module homomorphism
• Kernel of a ring homomorphism
• Algebraic relations between polynomials
• Hilbert polynomial of graded ideals and modules

Four examples

1) The hypersurface V(x2+y3-t2y2)

2) The variety V(xz, yz)

3) The space curve V(xz, xy, yz)

4) The set of points V(y4-y2, xy3-xy, x3y-xy, x4-x2)

The next questions and problems lead to algorithms which are slightly more (some of them much more) involved. They are, nevertheless, still very basic and quite natural. I should like to illustrate them by means of four simple examples, shown in the pictures of this section, referred to as Example 1) - 4):

Assume we are given an ideal $I \subset K[x_1, \dots, x_n]$ by a set of generators $f_1, \dots, f_k \in K[x]$. Consider the following questions and problems:

1. Is $V(I)$ irreducible or may it be decomposed into several algebraic varieties? If so, find its irreducible components. Algebraically this means to compute a primary decomposition of $I$ or of $\sqrt{I}$, the latter means to compute the associated prime ideals of $I$.

   Example 1) is irreducible, Example 2) has two components (one of dimension 2 and one of dimension 1), Example 3) has three (one-dimensional) and Example 4) has nine (zero-dimensional) components.

2. Is $I$ a radical ideal (that is, $I = \sqrt{I})$? If not, compute its radical $\sqrt{I}$.

   In Examples 1) - 3) $I$ is radical while in Example 4) $\sqrt{I} = \langle y^3 -y, x^3-x\rangle$, which is much simpler than $I$. In this example the central point corresponds to $V(\langle x,y\rangle^2)$ which is a fat point, that is, it is a solution of $I$ of multiplicity ( $= \dim_K K[x,y]/\langle x,y\rangle^2$) bigger than 1 (equal to 3). All other points have multiplicity 1, hence the total number of solutions (counted with multiplicity) is 11. This is a typical example of the kind Buchberger (resp. Gröbner) had in mind at the time of writing his thesis.

3. A natural question to ask is ``how independent are the generators $f_1, \dots, f_k$ of $I$?'', that is, we ask for all relations
   $$(r_1, \dots, r_k) \in K[x]^k,$$    such that $$\sum r_i f_i = 0.$$

   These relations form a submodule of $K[x]^k$, which is called the syzygy module of $f_1, \dots, f_k$ and is denoted by syz$\,(I)$. It is the kernel of the $K[x]$-linear map

   $$K[x]^k \longrightarrow K[x];\;\; (r_1, \dots, r_k) \longmapsto \sum r_i f_i.$$

4. More generally, we may ask for generators of the kernel of a $K[x]$-linear map $K[x]^r \longrightarrow K[x]^s$ or, in other words, for solutions of a system of linear equations over $K[x]$.

   A direct geometric interpretation of syzygies is not so clear, but there are instances where properties of syzygies have important geometric consequences cf. Sch3.

   In Example 1) we have syz$\,(I) = 0$, in Example 2), syz$\,(I) = \langle(-y,x)\rangle \subset K[x]^2$, in Example 3), syz$\,(I) = \langle(-z,y,0),(-z,0,x)\rangle \subset K[x]^3$ and in Example 4), syz$\,(I)\subset K[x]^4$ is generated by $(x,-y,0,0), (0,0,x,-y),(0,x^2-1,-y^2+1,0)$.

5. A more geometric question is the following. Let $V(I') \subset V(I)$ be a subvariety. How can we describe $V(I) \smallsetminus V(I')$? Algebraically, this amounts to finding generators for the ideal quotient
   $$I : I' = \{f \in K[x] \mid f I' \subset I\}.$$
   (The same definition applies if $I, I^\prime$ are submodules of $K[x]^k$.)

   Geometrically, $V(I : I')$ is the smallest variety containing $V(I) \smallsetminus V(I')$ which is the (Zariski) closure of $V(I) \smallsetminus V(I')$.

   In Example 2) we have $\langle xz, yz\rangle : \langle x,y\rangle = z$ and in Example 3) $\langle xy, xz, yz\rangle : \langle x,y\rangle = \langle z,xy\rangle$, which gives, in both cases, equations for the complement of the $z$-axis $x = y = 0$. In Example 4) we get $I : \langle x,y\rangle^2 = \langle y(y^2-1), x(x^2-1), (x^2-1) (y^2-1)\rangle$ which is the zero set of the eight points $V(I)$ with centre removed.

6. Geometrically important is the projection of a variety $V(I) \subset K^n$ into a linear subspace $K^{n-r}$. Given generators $f_1, \dots, f_k$ of $I$, we want to find generators for the (closure of the) image of $V(I)$ in $K^{n-r} = \{x\vert x_1 = \dots = x_r = 0\}$. The image is defined by the ideal $I \cap K[x_{r+1}, \dots, x_n]$ and finding generators for this intersection is known as eliminating $x_1, \dots, x_r$ from $f_1, \dots, f_k$.

   Projecting the varieties of Examples 1) - 3) to the $(x,y)$-plane is, in the first two cases, surjective and in the third case it gives the two coordinate axes in the $(x,y)$-plane. This corresponds to the fact that the intersection with $K[x,y]$ of the first two ideals is 0, while the third one is $xy$.

   Projecting the 9 points of Example 4) to the $x$-axis we get, by eliminating $y$, the polynomial $x^2 (x-1) (x+1)$, describing the three image points. From a set theoretical point of view this is nice, however it is not satisfactory if we wish to count multiplicities. For example, the two border points are the image of three points each, hence should appear with multiplicity three. That this is not the case can be explained by the fact that elimination computes the annihilator ideal of $K[x,y]/I$ considered as $K[x]$-module (and not the Fitting ideal). This is related to the well-known fact that elimination is not compatible with base change.

7. Another problem is related to the Riemann singularity removable theorem, which states that a function on a complex manifold, which is holomorphic and bounded outside a sub-variety of codimension 1, is actually holomorphic everywhere. This is well-known for open subsets of ${\mathbb{C}}$, but in higher dimension there exists a second singularity removable theorem, which states that a function, which is holomorphic outside a sub-variety of codimension 2 (no assumption on boundedness), is holomorphic everywhere.

   For singular complex varieties this is not true in general, but those for which the two removable theorems hold are called normal. Moreover, each reduced variety has a normalisation and there is a morphism with finite fibres from the normalisation to the variety, which is an isomorphism outside the singular locus.

   The problem is, given a variety $V(I) \subset K^n$, find a normal variety $V(J) \subset K^m$ and a polynomial map $K^m \longrightarrow K^n$ inducing the normalisation map $V(J) \longrightarrow V(I)$.

   The problem can be reduced to irreducible varieties (but need not be, as we shall see) and then the equivalent algebraic problem is to find the normalisation of $K[x_1, \dots, x_n]/I$, that is the integral closure of $K[x]/I$ in the quotient field of $K[x]/I$ and present this ring as an affine ring $K[y_1, \dots, y_m]/J$ for some $m$ and $J$.

   For Examples 1) - 4) it can be shown that the normalisation of the first three varieties is smooth, the last two are the disjoint union of the (smooth) components. The corresponding rings are $K[x_1, x_2],\; K[x_1, x_2] \oplus K[x_3],\; K[x_1] \oplus K[x_2] \oplus K[x_3]$. The fourth example has no normalisation as it is not reduced.

   A related problem is to find, for a non-normal variety $V$, an ideal $H$ such that $V(H)$ is the non-normal locus of $V$. The normalisation algorithm described below solves also this problem.

   In the examples, the non-normal locus is equal to the singular locus.

8. The significance of singularities appears not only in the normalisation problem. The study of singularities is also called local algebraic geometry and belongs to the basic tasks of algebraic geometry. Nowadays, singularity theory is a whole subject on its own.

   A singularity of a variety is a point which has no neighbourhood in which the Jacobian matrix of the generators has constant rank.

   In Example 1) the whole $t$-axis is singular, in the three other examples only the origin.

   One task is to compute generators for the ideal of the singular locus, which is itself a variety. This is just done by computing sub-determinants of the Jacobian matrix, if there are no components of different dimensions. In general, however, we need, additionally, to compute either the equidimensional part and ideal quotients or a primary decomposition.

   In Examples 1) - 4), the singular locus is given by $\langle x,y\rangle,\; \langle x,y,z\rangle,\;\langle x,y,z\rangle,\; \langle x,y\rangle^2$, respectively.

9. Studying a variety $V(I)$, $I = (f_1, \dots, f_k)$, locally at a singular point, say the origin of $K^n$, means studying the ideal $IK[x]_{\langle x\rangle}$ generated by $I$ in the local ring
   $$K[x]_{\langle x\rangle} = \left\{\dfrac{f}{g} \mid f,g \in K[x], g \not\in \langle x_1, \dots, x_n\rangle\right\}.$$
   In this local ring the polynomials $g$ with $g(0) \not= 0$ are units and $K[x]$ is a subring of $K[x]_{\langle x\rangle}$.

   Now all the problems we considered above can be formulated for ideals in $K[x]_{\langle x\rangle}$ and modules over $K[x]_{\langle x\rangle}$ instead of $K[x]$.

   The geometric problems should be interpreted as properties of the variety in a neighbourhood of the origin, or more generally, the given point.

It should not be surprising that all the above problems have algorithmic and computational solutions, which use, at some place, Gröbner basis methods. Moreover, algorithms for most of these have been implemented quite efficiently in several computer algebra systems, such as CoCoA, cf. CNR, Macaulay2, cf. GS and SINGULAR, cf. GPS, the latter also being able to handle, in addition, local questions systematically.

The most complicated problem is the primary decomposition, the latest achievement is the normalisation, both being implemented in SINGULAR.

At first glance, it seems that computation in the localisation $K[x]_{\langle x\rangle}$ requires computation with rational functions. It is an important fact that this is not necessary, but that basically the same algorithms which were developed for $K[x]$ can be used for $K[x]_{\langle x\rangle}$. This is achieved by the choice of a special ordering on the monomials of $K[x]$ where, loosely speaking, the monomials of lower degree are considered to be bigger.

However, such orderings are no longer well-orderings and the classical Buchberger algorithm would not terminate. Mora discovered, cf. Mo, that a different normal form algorithm, or, equivalently, a different division with remainders, leads to termination. Thus, Buchberger's algorithm with Mora's normal form is able to compute in $K[x]_{\langle x\rangle}$ without denominators.

Several algorithms for $K[x]$ use elimination of (some auxiliary extra) variables. But variables to be eliminated have, necessarily, to be well-ordered. Hence, to be able to apply the full power of Gröbner basis methods also for the local ring $K[x]_{\langle x\rangle}$, we need mixed orders, where the monomial ordering restricted to some variables is not a well-ordering, while restricted to other variables it is. In GP1 and Getal, the authors described a modification of Mora's normal form, which terminates for mixed ordering, and more generally, for any monomial ordering which is compatible with the natural semigroup structure.