In algebraic geometry, a line complex is a set of lines that can be specified by a list of homogeneous polynomial equations. That is, a projective variety of lines.

A linear line complex is defined by a list of degree-1 polynomials. A quadratic line complex is defined by a list of degree-2 polynomials. Similarly for cubic, quartic, quintic, sextic, etc.

They were first studied by Julius Plücker in Neue Geometrie des Raumes (1868). Other important figures include Felix Klein, Sophus Lie, Arthur Cayley, William Hamilton, and Alfred Clebsch.

Setup

By the standard trick in projective geometry, which is the Grassmannian via exterior product.

: the lines at infinity.

is the projectivized space of bivectors is the exterior product. The space has homogeneous coordinates (Plücker coordinates.

has 4 dimensions.

, but only a subvariety defined by the Klein quadric.

, using the embedding.

Higher dimensions

. This is called a sygyzy.

, then the other two conditions are automatically satisfied. This is the redundancy.

Special families of lines

A ruled surface.

orthogonal to a given surface. The Malus–Dupin theorem shows that geometric optics preserves orthogonal congruences.

Linear line complex

, which would be defined by a homogeneous linear equation which make a 2-form equal to zero. parameterization of the line. Equivalently, these are lines that are tangent to the cylindrical contact structureThat is, they are linear Legendrian submanifolds.

is a symplectic form, These planes are known as the isotropic planes.

Symplectic transformation

, then it is a symplectic transformation. In the spirit of Erlangen program, symplectic geometry studies invariants of symplectic transformations.

Symplectic transformations make up a group, called the symplectic group is a symplectic transformation. This is why Lie always referred to the symplectic group as "the projective group of a linear line complex".

Degeneracy

can be nondegenerate or degenerate. The symplectic form is nondegenerate.

.

special its axis.

Involution

Define the bilinear form are in involution, or in Klein polarity are orthogonal complements.

Pencils

.

More generally, the set of all lines through this point within a k-flat is embedded to a (k-1).

. This can be regarded as a planar pencil of a point at infinity.

More generally, the set of all lines within a k-flat is embedded to a (2k-2).

Constraints

In general, the set of all lines that satisfy a number of linear constraints is a linear line complex.

, the linear constraints include: passing a point, intersecting a line, and parallel to a plane. The set of all lines that intersect 2 lines and parallel to a plane (in general position) is a doubly ruled surface defined by intersecting 3 linear line complexes. In general, the set of all lines that satisfy 3 linear constraints is a regulus defined by intersecting 3 linear line complexes.

.

Quadratic line complex

symmetric matrix.

, a general quadratic line complex is specified by a quadratic form parameters are necessary. :

• Intersect a conic section.

• Tangent to a quadratic surface. In particular, fixed distance from a point (i.e. tangent to a sphere) and fixed distance from a line (i.e. tangent to a cylinder).

• Parallel to exactly line in a nondegenerate quadratic cone (i.e. a cone whose cross-section is a conic section). In particular, all lines that make a fixed angle to a plane (i.e. parallel to a line in a circular cone). Equivalently, all lines that make a fixed skew-angle to a line.

• Given two quadratic surfaces and a real number, the lines that intersect the two surfaces at 4 points whose cross-ratio forms a quadratic line complex. These are the harmonic complexes.

The intersection of 3 linear/quadratic constraints in general position produces a ruled surface. For example, the tangent developable of a helix is obtained by intersecting the 3 constraints of: tangent to a cylinder, makes a fixed angle with a plane, and in the Legendrian line complex. Thus it is a quadratic line complex.

Tetrahedral

In the special case where the two quadratic surfaces degenerate into 4 planes, the lines that intersect the 4 planes at 4 points whose cross-ratio is equal to a fixed number forms a quadratic line complex. This is a tetrahedral complex, which is a special case of the harmonic complex.

von Staudt.

For any quadratic complex and linear complex in general position, their intersection is contained in a tetrahedral complex.

Singular

In general, a quadratic line complex defined by a single quadratic equation there exists a whole 1-dimensional pencil of lines that sweeps out a quadratic cone. This can be constructed by intersecting the quadratic constraint with the linear constraint of "passing to a point".

However, there exists a 2-dimensional set of points where this breaks down. Specifically, at these points, the quadratic cone degenerates into two flat planes. These points are the singular points of the quadratic line complex. The set of these points is its singular surface. The singular surface is Kummer's surface. It is a quartic surface. Each line meets the surface in 4 points, and through each line there are 4 tangent planes to the surface. It has 16 singular points ("nodes") and 16 singular planes ("tropes"), the maximum possible for a quartic surface. There are 6 nodes lying in each of the 16 tropes and 6 tropes passing through each of the 16 nodes.

intersects a quadratic line complex at a 1-dimensional quadratic line complex. In general, these lines sweep out a conic section as their envelope. However, there exists a 2-dimensional set of planes where this breaks down. These planes are the singular planes. The envelope of these planes is once again the same singular surface.

There is a 1-dimensional family of infinite number of quadratic complexes having the same singular surface. These are called cosingular complexes.

Higher line complex

. They can be obtained by intersecting 3 constraints in general position. The only structurally stable singularity that can appear in a ruled surface is the cross-cap, which looks like a Whitney unbrella.

.

In particular, the intersection of a quadratic complex with a linear complex is a congruence with order 2 and class 2.

History

At the death of Julius Plücker, his Neue Geometrie des Raumes was completed by his pupil Felix Klein in 1868. Klein used invariant theory that he learned from Alfred Clebsch.

Klein and Lie studied line complexes, especially tetrahedral complexes. The group of all projective transformations that stabilizes the 4 vertices of the tetrahedron is a 3-dimensional abelian Lie group. Each tetrahedral complex is obtained as the orbit.

. He called these the W-curves. Since the curve is essentially translated by the action of the group, it necessarily maps tangents to tangents. This was an application of Lie algebra in the prehistory of Lie group theory. The investigation of such linear complexes also inspired Klein to propose the Erlangen program.

Their line of work in line geometry was taken up by Corrado Segre, whose work, including the classification of quadratic line complexes, was foundation in the Italian school of algebraic geometry.

References