Figures 2.1 and 2.2 show two geometric tests of 2-dependence, one based on a conic curve and the other based on a complete quadrilateral. In both cases, the six scalars being tested are encoded as the slopes of six lines through a common point. Recall that duality in the projective plane interchanges the concepts of ‘point’ and ‘line’. The duals of the situations in Figures 2.1 and 2.2 — shown in Figures 2.5 and 2.6 — are also geometric tests of 2-dependence. But the six scalars being tested in Figures 2.5 and 2.6 are the coordinates of six points along a common line. In Figure 2.5, we have an unordered 2-block {A1,B1},{A2,B2},{A3,B3} of points, all lying on a common line o. There is a conic k of which o is a tan- gent line and there are three collinear points H1, H2, and H3 with the property that the tangents ai and bi from Hi to the conic k intersect o at Ai and Bi. In Fig-
2.6. THE DUAL POINT OF VIEW 23 o A1 A2 A3 B1 B2 B3 A Q R S
Figure 2.6: The same 2-block of points along the same line o as in the previous fig- ure, but here demonstrated to be a quadrangular set by the existence of the complete quadrangle with vertices(A,Q,R,S).
ure 2.6, we interpret the same six points along the same line o as an ordered 2-block {(A1,B1), (A2,B2), (A3,B3)}. There are four auxiliary points A, Q, R, and S with the property that the points{(Ai,Bi)}i∈[1..3]are located where the line o stabs the lines
a1 := AQ b1 := RS
a2 := AR b2 := Q S
a3 := AS b3 := Q R.
The four auxiliary points and the three pairs of opposite lines joining them are called a complete quadrangle.4 Three pairs of points on a common line have the conic-based property of Figure 2.5 if and only, as in Figure 2.6, they are a cross section of a complete quadrangle, and they are then called a quadrangular set.
Figures 2.5 and 2.6 geometrically characterize the same algebraic notion of 2-dependence that is characterized in Figures 2.1 and 2.2: If we choose any projec- tive coordinate system on the line o, the six points{A1,B1},{A2,B2},{A3,B3} form a quadrangular set just when their coordinates along o form a 2-dependent block of scalars. Thus, it makes no essential difference which of the dual geomet- ric situations we discuss. The situation in Figures 2.5 and 2.6 is the one generally presented in textbooks. For our purposes, however, it is more convenient to talk about the situation in Figures 2.1 and 2.2.
Why is that? Our goal is to study projective configurations that characterize
n-dependence, which means that we have to make a choice. As in Figure 2.2, we
can test the n-dependence of the slopes of an n-block of hyperplanes by requir- ing that those hyperplanes result from projecting the points of some instance of a configuration C. Of course, the configuration C will involve various flat sub- spaces of various dimensions; but it must involve an n-block of points, and we
4Indeed, Figure 2.6 is quite similar to Figure 0.1, except that the point P has been relabeled A,
the line m has been relabeled o, and the separation properties — as discussed in Exercise 2.1-3 — of the pairs{Ai,Bi}are different. In Figure 2.6, each pair separates both of the other pairs, while,
24 CHAPTER 2. INTRODUCTION
are likely to think of its other flats as built up by joining certain subsets of those points. Alternatively, as in Figure 2.6, we can test the n-dependence of the coordi- nates of an n-block of points by requiring that those points result from stabbing the hyperplanes of some instance of the dual configuration C∗. The configuration C∗ also involves various flats of various dimensions; but it must involve an n-block of hyperplanes, and we are likely to think of its other flats as formed by intersect- ing certain subsets of those hyperplanes. The two situations are dual, and hence completely equivalent. But it is somewhat simpler to talk about building a config- uration by working up, starting from an n-block of points, rather than by working down, starting from an n-block of hyperplanes. This is especially true when ma- troids are involved, since the tradition is to represent a matroid by mapping the elements of its ground set to points, rather than to hyperplanes.
Don’t get confused between the duality of projective geometry and the duality of matroid theory. The duality of projective geometry interchanges points and hy- perplanes, as we have been discussing. The duality of matroid theory associates, with each matroid of rank r on s points, a dual matroid of rank s−r on those same s points. We are going to define a family of matroids, and each of those matroids
does have a dual. But we shall have no further occasion to mention those dual ma- troids in this monograph. As that fact suggests, we aren’t going to be appealing to any deep results of matroid theory; instead, we use matroids merely as a formal framework in which to talk about projective configurations.
Exercise 2.6-1 Given a 3-block{αi, βi, γi} i∈[1..4]of planes in 3-space, all pass- ing through a common line o, we discussed in Section 2.2 how to test the resulting block of slopes for 3-dependence geometrically, using a twisted cubic curve. Du- ally, given a 3-block{Ai,Bi,Ci} i∈[1..4]of points in 3-space, all lying on a com- mon line o, describe how to test the resulting block of coordinates for 3-depen- dence geometrically, using a twisted cubic.
[Hint: Choose your twisted cubic so that o is the line where two osculating planes intersect.]