When is it that six points form an instance of the complete quadrilateral? Any four of the six points must be coplanar and certain triples of them must be collinear. In
2.8. MATROIDS 27 addition, because we have agreed to forbid every incidence that is not required, the remaining triples must not be collinear and no two of the six points may coincide.
Thus, we can think of the complete quadrilateral as a set of abstract points and a rule that says, for each subset of those points, whether the k points in that subset must or must not lie in a common(k−2)-flat.5 In this monograph, we shall refer to any such rule as a projective configuration. Warning: In classical projective ge-ometry, such a rule has to have lots of other properties, including a high degree of numeric symmetry, before it can be called a configuration. We discuss that narrow sense of the word ‘configuration’ in Section 4.5.
In describing the incidences that are required by a configuration, there is no need to use separate terms, such as ‘collinear’ and ‘coplanar’, for sets of different sizes; the single term ‘incident’ will do for all. Call k points in some projective space mutually incident when they lie in a common(k −2)-flat. Algebraically, this means that the k vectors of homogeneous coordinates are linearly dependent.
So four points are mutually incident when they are coplanar; three points, when they are collinear; two points, when they coincide; and one point, when it is inde-terminate — that is, all of its homogeneous coordinates are equal to zero.6 When k points are not mutually incident, we shall call them mutually skew.
Note that every superset of a mutually incident set is also mutually incident; for example, if the three points A, B, and C are collinear, then the four points A, B, C, and D must be coplanar. To make this property hold also when the smaller set is empty, we make the convention that the zero points in the empty set∅are mutually skew, rather than mutually incident; this agrees with the standard convention that the empty set of vectors is linearly independent.
An instance of a configuration in some projective space maps each abstract point of the configuration to a concrete point in that projective space so that pre-cisely the required incidences hold. Note that, since we have defined the notion of a configuration quite broadly, it is easy to come up with configurations that have no instances. For example, one way to guarantee that no instances exist is to fool-ishly require that the set of points S be mutually incident while forbidding the set T from being mutually incident, for some T ⊃ S. Another way is to foolishly require that the empty set∅be mutually incident.
We are mostly interested in configurations that do have instances. Hence, when
5We use the term ‘m-flat’ to mean a flat subspace of dimension m; thus, a line is a 1-flat and a plane is a 2-flat. Be warned that many authors writing about matroids use ‘m-flat’ to mean a flat of rank m, which has dimension m−1.
6It is often desirable, as Stolfi [51] points out, to augment projective space with a unique null object of each dimension. The traditional names ‘indeterminate point’, ‘indeterminate line’, and so forth for these null objects have the advantage that they encode the dimension in a very natural way.
Using the term ‘indeterminate point’ has the severe disadvantage of making it forever afterwards unclear, when we assume that P is a point in a projective space, whether we are allowing P to be indeterminate or not. In this monograph, fortunately, the indeterminate point arises only while we are reviewing matroids and their representations. It never arises thereafter because the particular matroids that we define have no single-element dependent sets — that is, no loops.
28 CHAPTER 2. INTRODUCTION designing our configurations, it behooves us to avoid the two sorts of foolishness just mentioned. There is a third, less obvious sort of foolishness that we should also avoid, and any configuration that avoids all three is called a matroid. More formally, as we review in Chapter 3, a matroid is a configuration that satisfies a certain three axioms. Those axioms are not strong enough to guarantee that every matroid has instances; but they do eliminate three sorts of foolishness, at least, and they thereby ensure that the combinatorial structure of the configuration has some nice properties. What we are calling an instance of a configuration is called, in ma-troid theory, a representation of the mama-troid. Not every mama-troid is representable;
but every configuration that has any instances is a representable matroid.
The cubic analog of the complete quadrilateral, which is the configuration for which we are searching, turns out to correspond to a particular representable ma-troid, which we shall call B2,1,1. As that fancy name suggests, the matroid B2,1,1 belongs to a family of matroids: the budget matroids, which we define in Chap-ter 4. There is a budget matroid Bb1,...,bk associated with each partition b = b1 +
· · · +bk of an integer b into nonnegative parts, at least two of which are positive.
The complete quadrilateral is the budget matroid B2,1 — and hence that matroid characterizes 2-dependence. In an analogous way, the matroid B2,1,1characterizes 3-dependence. Unfortunately, this pattern doesn’t continue: The matroid B2,1,1,1 characterizes some property that is stronger than 4-dependence.
Each of the parts bj in the partition b = b1 + · · · + bk that determines the budget matroid Bb1,...,bk actually plays a double role. From the point of view of matroid theory, there is no reason not to separate those two roles, assigning them to independent parameters bj and dj. Chapter 5 does this, thereby defining a larger family of matroids, which we call the budgetary matroids. From a geometric point of view, however, this generalization seems to be of little value. Indeed, many of the budgetary matroids turn out not to be representable at all.
The representability of the budget matroids makes for a happier story. If the budget matroid Bb1,...,bk is representable, it is fairly easy to see that the budget ma-troids that result from reducing or eliminating one or more parts in the partition b=b1+ · · · +bk are also representable. So, in proving representability, the easy cases are those with few parts and with small parts. In Chapter 6, we show that ev-ery budget matroid of the form Bm,n, with precisely two parts, is representable over the rational numbers and hence over every field of characteristic zero. In Chapter 7, we show the same result for the budget matroids of the form Bm,1,1. Neither proof is very difficult, but both are fairly long. The latter proof involves a theorem that can be interpreted as generalizing Pappus’s Theorem to higher dimensions.
Since the budget matroid Bm,1,1is representable for every positive m, the par-ticular matroid B2,1,1is representable. Unfortunately, the scheme that we develop for constructing representations of the matroids Bm,1,1in Chapter 7 doesn’t seem to be helpful in studying the relationship between the matroid B2,1,1and the concept of 3-dependency. To get a handle on that relationship, we need a different con-struction, one that makes its choices in a different order. That second construction
2.9. NULL SYSTEMS 29