Line complex

 ( Algebraic Varieties ) 

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.

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Setup By the standard trick in projective geometry, a line in 3-dimensional space is lifted to a plane through the origin in 4-dimensional space. In other words, the space of lines in $\backslash R^3$ is lifted to the space of planes through the origin in $\backslash R^4$, which is the Grassmannian $G(2,\; 4)$. It is then embedded to the projective space $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$ via exterior product.

Note that during the projective embedding, we get lines that does not exist $\backslash R^3$: the lines at infinity.

$\backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$ is the projectivized space of Bivector in $\backslash R^4$, where $\backslash wedge$ is the exterior product. The space has homogeneous coordinates (Plücker coordinates) $p\_\{12\},$. By convention, if $p\_\{ij\}$ where $i\; >\; j$ is written, then $p\_\{ij\}\; :=\; -p\_\{ji\}$.

$\backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$ has 5 dimensions, and $G(2,\; 4)$ has 4 dimensions.

A plane in $G(2,\; 4)$ can be described by any two nonparallel vectors $v,\; w$, then it is mapped to $v$ with coordinates $p\_\{ij\}\; =\; v\_iw\_j\; -\; v\_j\; w\_i$. Thus we obtain an embedding $p\; :\; G(2,\; 4)\; \backslash hookrightarrow\; \backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$. It is not the whole $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$, but only a subvariety defined by the Klein quadric:$$p\_\{12\}\; p\_\{34\}+p\_\{13\}\; p\_\{42\}+p\_\{14\}\; p\_\{23\}\; =\; 0$$More succinctly, $p\; \backslash wedge\; p\; =\; 0$. Note that this is not a trivial condition, as a generic bivector does not satisfy it, for example $(e\_1\; \backslash wedge\; e\_2\; +\; e\_3\; \backslash wedge\; e\_4)$.

From now on we will simply write $G(2,\; 4)$ as if it is a subset of $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$, using the embedding.

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Higher dimensions The same construction shows that the space of lines in $\backslash R^n$ is lifted to $G(2,\; n+1)$, then embedded as a subvariety of $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$. Since $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$ has $\backslash tfrac\; 12\; n(n-1)-1$ dimensions, and $G(2,\; n+1)$ has $2(n-2)$ dimensions, it should be possible to write it as the intersection of $\backslash tfrac\; 12\; (n-2)(n-3)$ hypersurfaces. However, $G(2,\; \backslash R^n)$ is the intersection of the $\backslash tfrac\; 14\; n(n-1)(n-2)(n-3)$ equations:Linear relation.

For example, when $n\; =\; 5$, the space $G(2,\; 5)$ is a codimension-3 subspace of $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^5)$ defined as the intersection of 5 equations, so there are 2 redundancies, which can be exhibited by the matrix equation:In the subspace of $\backslash mathbb\; A^9\; \backslash subset\; \backslash mathbb\; P(\backslash wedge^2\; \backslash R^5)$ where $p\_\{12\}\; =\; 1$, the two linear syzygies$$Q\_1=p\_\{23\}\; Q\_3+p\_\{24\}\; Q\_4+p\_\{25\}\; Q\_5,\; \backslash quad\; Q\_2=-p\_\{13\}\; Q\_3-p\_\{14\}\; Q\_4-p\_\{15\}\; Q\_5$$show that if $Q\_3,\; Q\_4,\; Q\_5\; =\; 0$, then the other two conditions are automatically satisfied. This is the redundancy.

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Special families of lines A ruled surface in $\backslash R^3$ is a curve in the Klein quadric $G(2,\; 4)\; \backslash subset\; \backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$. A ruled surface is a line complex when it is the intersection of 3 line complexes, each defined by a single homogeneous polynomial on $\backslash wedge^2\; \backslash R^4$.

In general, a ruled surface is a 1-parameter family of lines. A congruence of lines is a 2-parameter family. A particular example is the set of lines in $\backslash R^3$ orthogonal to a given surface. The Malus–Dupin theorem shows that geometric optics preserves orthogonal congruences.

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Linear line complex A linear line complex is the intersection of $G(2,\; 4)$ with one or more hyperplanes in $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^4)$, which would be defined by a homogeneous linear equation. Equivalently, it is defined as the space of planes through the origin in $\backslash R^4$ which make a 2-form in $\backslash wedge^2\; \backslash R^4$ equal to zero. For example, $p\_\{12\}\; +\; p\_\{34\}\; =\; 0$ defines the space of planes through the origin in $\backslash R^4$ which make $\backslash omega\; :=\; dx\_1\; \backslash wedge\; dx\_2\; +\; dx\_3\; \backslash wedge\; dx\_4$ equal to 0. There are $\backslash infty^3$ such planes. Equivalently, it is a 3-dimensional space of lines in $\backslash R^3$ (plus the lines at infinity), which satisfy $v\_2\; w\_1\; -\; v\_1\; w\_2\; +\; w\_3\; =\; 0$ given any $\backslash \{v\; +\; t\; w\; :\; t\; \backslash in\; \backslash R\backslash \}$ parameterization of the line. Equivalently, these are lines that are tangent to the cylindrical contact structure on $\backslash R^3$:$$dz\; -xdy\; +\; ydx\; =\; dz\; -\; r^2\; d\backslash theta$$That is, they are linear Legendrian submanifolds.

More generally, $p\_\{12\}\; +\; \backslash dots\; +\; p\_\{2n-1,\; 2n\}\; =\; 0$ defines the space of planes through the origin in $\backslash R^\{2n\}$ which make $\backslash omega\; :=\; dx\_1\; \backslash wedge\; dx\_2\; +\; \backslash dots\; +\; dx\_\{2n-1\}\; \backslash wedge\; dx\_\{2n\}$ equal to 0. There are $\backslash infty^\{2n\; -5\}$ such planes. This form $\backslash omega$ is a symplectic form, These planes are known as the isotropic planes of $\backslash omega$.

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Symplectic transformation Any linear transformation of $\backslash R^n$ induces a transformation of $\backslash wedge^2\; \backslash R^\{n+1\}$ that preserves $G(2,\; n+1)$. If the transformation also preserves $\backslash omega$, 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. The group acts on $\backslash R^n$, splitting it into 2 orbits: one orbit for nonzero vectors, and another orbit for 0. It acts on $G(2,\; n+1)$ as well, splitting it into 2 orbits: one orbit for isotropic planes, and another orbit for symplectic planes. More generally, any projective transformation on $\backslash mathbb\; P(\backslash R^n)$ that preserves the linear line complex defined by $\backslash omega$ is a symplectic transformation. This is why Lie always referred to the symplectic group as "the projective group of a linear line complex".

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Degeneracy A 2-form $\backslash sigma$ on $\backslash R^\{n+1\}$ can be Non-degenerate or degenerate. The symplectic form is nondegenerate.

In the degenerate case, its kernel is a nontrivial even-dimensional linear subspace: $\backslash ker\; \backslash sigma:=\; \backslash \{v\; \backslash in\; \backslash R^\{n+1\}\; |\; \backslash sigma(v,\; \backslash cdot)\; =\; 0\backslash \}\; \backslash subset\; \backslash R^\{n+1\}$, which projectives to an odd-dimensional flat in $\backslash mathbb\; P(\backslash R^\{n+1\})$.

A degenerate but nonzero 2-form on $\backslash R^4$ has a 2-dimensional kernel, which projectivizes to a line in $\backslash R^3$. Plücker called a linear line complex defined by a degenerate $\backslash sigma$ special, and $\backslash ker\; \backslash sigma$ its axis.

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Involution Define the bilinear form on $\backslash wedge^2\; \backslash R^n$. Two linear line complexes defined by two linear subspaces $V,\; W\; \backslash subset\; \backslash wedge^2\; \backslash R^n$ are in involution, or in Klein polarity, if $V,\; W$ are orthogonal complements relative to $B$.

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Pencils Through a point in $\backslash R^n$, a planar pencil is the set of all lines through this point within a 2-dimensional plane. It is embedded to a straight line in $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$.

More generally, the set of all lines through this point within a k-flat is embedded to a (k-1)-flat in $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$.

Given a 2-flat $F\; \backslash subset\; \backslash R^n$, the set of all lines in it is enumerated by the pencils of each point in a line. It is embedded to a 2-flat in $p(F)\; \backslash subset\; \backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$. This can be regarded as a planar pencil of a point at infinity. Dually, for each $p(l)\; \backslash in\; p(F)$, there is a planar pencil of lines in $p(F)$. Each line in the pencil of $p(l)$ corresponds to a pencil in a point in $l\; \backslash subset\; F$.

More generally, the set of all lines within a k-flat is embedded to a (2k-2)-flat in $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$.

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

In $\backslash R^3$, 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.

More exotic examples are available. For example, the aforementioned family of Legendrian lines defined by $p\_\{12\}\; +\; p\_\{34\}\; =\; 0$ is a linear line complex, which can be interpreted as a linear constraint of $dz\; -xdy\; +\; ydx$.

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Quadratic line complex A quadratic line complex is obtained by intersecting $G(2,\; n+1)$ with quadratic hypersurfaces in $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^\{n+1\})$. In homogeneous coordinates, they are of form $p^T\; M\; p$, where $M$ is a $\backslash tfrac\; 12\; n(n+1)\backslash times\backslash tfrac\; 12\; n(n+1)$ symmetric matrix.

In the case of $n=3$, a general quadratic line complex is specified by a quadratic form on $\backslash mathbb\; P(\backslash wedge^2\; \backslash R^3)$ modulo the Klein quadric, resulting in 19 parameters. In general, $\backslash binom\{\backslash tfrac\; 12\; n(n+1)\}\{2\}\; -\; \backslash binom\{n+1\}\{4\}-\; 1$ parameters are necessary. In general, the set of all lines that satisfy a number of linear constraints, and at least one quadratic constraint is a quadratic line complex. Examples in $\backslash R^3$:

• Intersect a conic section.
• Tangent to a Quadric. 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 $c$, the lines that intersect the two surfaces at 4 points whose cross-ratio is equal to $c$ 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.

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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's theorem: given a line $l$ in general position with a tetrahedron with vertices $ABCD$ and their respective faces $abcd$, the cross-ratio of the 4 planes $lA,$ is equal to the cross-ratio of the 4 points $l\backslash cap$.

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

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Singular In general, a quadratic line complex defined by a single quadratic equation has 3 dimensions, so through each point in $\backslash R^3$ 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 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.

Similarly, each plane in $\backslash R^3$ 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.

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Higher line complex A ruled surface is a 1-parameter family of lines $\backslash R^3$. 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 umbrella.

A congruence of lines is a 2-parameter family in $\backslash R^3$, and are obtained by intersecting 2 constraints in general position. In general, through each point in $\backslash R^3$ there passes exactly $n$ lines of the congruence, and through each plane in $\backslash R^3$ there passes exactly $m$ lines of the congruence. Such a congruence is said to be of order $n$ and class $m$.

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

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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 group Lie group $G$. For any two points $x,\; y$ in general position with the tetrahedron, there exists a unique $g\; \backslash in\; G$ such that $g(x)\; =\; y$. Each tetrahedral complex is obtained as the orbit of a single line under the action of $G$.

Lie asked Klein whether there are curves $\backslash gamma$ such that, if $g\backslash in\; G$ maps some $x\; \backslash in\; \backslash gamma$ to some $g(x)\; \backslash in\; \backslash gamma$, then $g$ also maps the tangent line $T\_x\; \backslash gamma$ to the tangent line $T\_\{g(x)\}\; \backslash gamma$. Klein discovered that there are curves which are stabilized by a 1-dimensional subgroup of $G$. He called these the W-curve. 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.

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