Plane Configurations of Lines

At first glance it might seem that the world of configurations of lines in a plane resembles the world of configurations of lines in three-dimensional space, which we made an attempt to understand above. It is certainly easy to give definitions for plane configurations which are analogous to the basic definitions in this article. But, contrary to our expectations, these two worlds have very little in common.

Undoubtedly, the plane configuration analog of an interlacing of skew lines is a configuration of lines no three of which pass through a point and no two of which are parallel. The analog of an isotopy of interlacings is a motion during which the lines remain lines and the conditions on the location of the lines are preserved.

Passing from the plane to the projective plane changes the problem, and here, as usual, the projective problem turns out to be simpler and more elegant. In the projective problem the objects are sets of projective lines in $\mathbb{R}P^3$ which satisfy only one condition: no three of them pass through a point. Such a projective plane configuration of lines will be said to be nonsingular. A configuration of this type can also be interpreted as a set of planes through the origin in $\mathbb{R}^3$ such that no three of them contain a line.

In the case of nonsingular plane configurations of lines one must distinguish between isotopies and rigid isotopies. Two configurations are isotopic, or, equivalently, they have the same topological type, if one can be taken to the other by means of a homeomorphism $\mathbb{R}P^2\to\mathbb{R}P^2$. Two configurations are said to be rigidly isotopic if they can be connected by a path in space whose points are nonsingular plane configurations of lines.

In the isotopic and rigid isotopic classification problems for plane configurations we do not have the amphicheirality question. This is because the mirror image of any configuration is isotopic to the original configuration, since a reflection of the projective plane is isotopic to the identity map by means of an isotopy consisting of projective transformations. (More generally, the group of projective transformations of $\mathbb{R}P^2$ is connected.)

The isotopic and rigid isotopic classification problems for nonsingular plane configurations of lines have both been solved for configurations where the number of lines is $\le7$, and in these cases the answer to both problems turns out to be the same (see Finashin []8). If there are $\le5$ lines, the isotopy type is determined by the number of lines. There are four types of nonsingular plane configurations of six lines, and 11 types of nonsingular plane configurations of seven lines. But when we reach configurations of more than seven lines, the isotopy and rigid isotopy classifications diverge sharply. Mnev []9 proved a surprising theorem, according to which, roughly speaking, a set of nonsingular plane configurations of lines which are isotopic to one another can have the homotopy type of any affine open semi-algebraic set, and, in particular, it can have any number of connected components, i.e., it can contain an arbitrary number of rigid isotopy classes. (This statement is imprecise, because in Mnev's work one considers ordered configurations in which the first four lines are in a fixed position; otherwise one must divide out by the action of the group of projective transformations.)

The simplest example known of nonsingular plane configurations of lines which are isotopic but not rigid isotopic can be found in Suvorov [13]. The configurations in this example have 14 lines.