Lines in space and the Grassmann variety

This is an English version of my previous post. Schubert calculus is a collection of techniques and formulae used for computations of enumerative, numerical properties of common objects in linear algebra (there are very nice books by William Fulton covering the subject). The example of the set of lines in three-dimensional space is commonly chosen to illustrate the kind of results which are obtained by Schubert calculus: the reason is probably that it is the simplest non trivial situation.

A line in affine space would usually by defined by its direction (which is a line in a vector space, and depends on two parameters: imagine parameterising it by a vector on a sphere), and its position, depending on two more parameters, for a given direction (think of the possible translations of a given line). Thus the set of lines might be parameterised by four parameters, and more precisely one can show that it has dimension four. Even more interestingly, two parallel planes define for almost every line a unique pair of intersection points (one on each plane) which enables us to recover the line. This means lines are (almost all) parameterised by four coordinates, by means of rational functions (well, we still have to explain what is parameterised: for example, we could say that the slope, direction of the line and its position are rational functions of the coordinates on the two planes). The resulting parameterisation is somewhat bijective, but a few lines are missing (for example, lines parallel to the plane): however, we say the set of lines is a rational variety: which means the parameterisation is essentially bijective, and the missing elements can be obtained by taking limits.

The Plücker embedding

To avoid tiring case disjunctions, it is very nice to extend the theory to the set of lines in projective space. They can be defined by their Plücker coordinates : given a line in space, we choose two points, given by projective (a.k.a. barycentric) coordinates and . Then we define the Plücker coordinates of the line by the six numbers . The fundamental properties of these coordinates is their invariance under changing the choice of points. Any other pair of points would be given by barycentres of M and N (with weights and ), giving new coordinates


which are actually proportional to the old coordinates. These coordinates, as projective coordinates, are thus uniquely defined (and even allow to recover the line). The set of lines now appears as a subvariety of a projective space called the Grassmann variety of lines in 3-space.

Remember that this variety is rational: we described a way to obtain almost every line using points and (chosen in planes and ). The resulting line has Plücker coordinates
, , , ,
,

Thus it is easy to see that the Grassmannian variety is defined by a degree 2 equation

which is a quadratic form: we say the Grassmann variety is a quadric. The latter parameterisation shows it looks like the graph of the polynomial .

Line on a hyperboloid

Given three lines in space chosen randomly enough, there is a unique one-sheeted hyperboloid (in short, a quadric) containing them. To see this, notice that we can choose projective coordinates such that one of the lines is described by whereas the other becomes the set of points (choose two points on each line as a basis for barycentric coordinates). Then any quadric containing these two lines has an equation of the form
(where x, y, z, t are the chosen coordinates).

In order to contain the third line, the quadric has to go through three points on it (this is a way of saying a degree two polynomial is zero if it vanishes three times). This translates into three linear equations for a, b, c, d, leaving only one possible equation up to scaling. A Gauss reduction of the equation has the form :

We say the equation has a quadratic form a signature (2,2): which geometrically means the corresponding surface is a one-sheeted hyperboloid. It is well known that such a surface is swept by two systems of lines (called rulings). To see this draw a curve around the hyperboloid and see that at each point the tangent plane to the hyperboloid cuts it along two lines going in opposite directions.

Examples of Schubert conditions

A Schubert condition for lines in space is a constraint belonging to the following list :

• : it goes through a given point,
• : it cuts a given line,
• : it is contained in a fixed plane,
• : it is constrained in a fixed plane and also goes through a fixed point,

The Schubert conditions are in a certain sense the elementary bricks used to distinguish a subset of the Grassmann variety of lines in space. The conditions defined by fixed points of planes are very easy to study, but the condition “cutting a given line” is trickier to understand. Schubert calculus gives tools to understand what lines satisfying several such conditions look like.

For example, suppose we are given two lines and in general position in space. Then from almost any point in space, these lines look like two lines from our point of view, crossing in the “sky” (a plane of projection). So there is usually a direction from our point of view which crosses the two lines, which defines a line crossing the two lines. The set of these lines is almost a partition of the three-dimension space into lines, which can be rationally parameterised by choosing points in a fixed plane as points of view. This, however, does not describe precisely the shape of this set of lines.

Schubert calculus gives a formula . This formula has a geometric interpretation in the case and are coplanar and secant: it is easy to check that any line crossing both of them is either in their plane or goes through their common point. The formula thus means: “satisfying two conditions is equivalent to satisfy or “.

Now if we are given three generic lines in space, we saw there was a unique hyperboloid containing all of them. A line satisfying simultaneously the three associated Schubert conditions can be shown to lie on the hyperboloid (the three fixed lines belong to the same ruling, and lines in different rulings intersect). These lines can be parameterised by a conic : if you cut the hyperboloid by a plane, each point of the resulting conic will define a line in each ruling and one of them cuts the three given lines.

Schubert calculus gives the formula , which makes sense in the following situation : suppose and are skew lines cutting in different points. Then a fourth line has two ways of cutting them all: either it goes through while sitting in the same plane as , or it goes through while being coplanar with (we recognize two conditions of type ). Part of this information remains relevant in the general situation: these two conditions are linear, thus defines (in an adequate sense) a degree two condition, which is also the case of the condition defined by a hyperboloid. The Schubert calculus does not distinguish these two possibilities.

If we now choose four generic lines in space, we can show there are exactly two lines cutting them all. To understand this, we can choose a hyperboloid contaning the first three lines (red in the figure), so that cutting three of the lines constrains our varying line to be in a ruling (black lines). The fourth Schubert condition forces the black line to go through one of the intersection points of the fourth line with the hyperboloid, so there are two solutions. This is formulated in Schubert calculus by . However, the statement is false, since there are lines not intersecting the hyperboloid at all!
The correct interpretation of this is that even if the fourth line seems not to intersect the hyperboloid, the corresponding equations should have two imaginary solutions, so there are two imaginary lines cutting the four given lines simultaneousl: Schubert calculus does not distinguish real or complex solutions.