Projective plane

Fano's plane is an example of a projective plane.

The projective plane is the set studied by projective geometry. It arises in Euclidean geometry by adding to a plane a point for each family of parallel lines (ie, one for each pair of opposite directions). The points thus added are called points of infinity, and their introduction greatly unifies and simplifies the statements of geometry. For example, the statement that says that two lines of a plane intersect at a single point or are parallel, in the projective plane it is stated: Two lines always intersect at a single point.

Use in projective geometry

Projective geometry really arises by studying "only" incidence relations when ordinary points and points of infinity are considered on an equal footing. One way to visualize projective geometry is to take a point P outside the plane and define it as the projections of three-dimensional elements onto a plane. Each point in the plane clearly defines a line through P; but thus we do not obtain all the lines that pass through P: the lines parallel to the given plane, which correspond precisely to the directions of the plane, are missing.

The points of the projective plane correspond naturally with the lines that pass through P, and the lines of the projective plane with the planes that pass through P. This correspondence preserves the incidence relations, it is an isomorphism between their respective structures. We can simply define the projective plane as the radiation of lines of vertex at a given point P. In addition, this perspective allows us to also introduce the projective line, as radiation of lines of given vertex in a plane, and three-dimensional projective spaces, as radiation of lines that they pass through the origin in a vector space of dimension 4, or of any other dimension n (just considering vector spaces of dimension n+1).

From the poverty of statements to their wealth

Seeing its two principles (two lines intersect at a point and two points define a line), projective geometry seems the poorest of all geometries, since only the concept of incidence intervenes in its statements. It does not admit the concepts of parallelism, perpendicularity, distance or angle. It only allows incidence statements such as:

• "For two points, a single one passes" or
• Theorem of Desargues: «If the three straights that unite the vertices of two triangles converge at a point, then the three cut points of the extensions of the corresponding sides are aligned».

However, as we have seen, the geometric statements in which the concept of parallelism also intervenes (the so-called affine geometry) can be reformulated in the projective plane without more than fixing a line, a line that then receives the name of «line of the infinite". Thus, every affine statement admits an equivalent projective statement, and affine geometry can be seen as a small part of projective geometry: it is the geometry of a projective plane with a prefixed line (or a plane in projective space, etc.). Likewise, it was seen that Euclidean geometry is obtained by fixing two conjugated complex points on the line of infinity (the points where all circumferences intersect), thus being included in Projective Geometry: it is the geometry of a projective plane where two complex conjugate points of a line. Even hyperbolic geometry, the first of the non-Euclidean geometries, can be obtained by fixing a conic: the points of the geometry are the interior points of the conic, the lines are the sections of the interior of the conic with lines, and the distance between two points A, B is essentially their double ratio with the intersection points P, Q of the line AB with the given conic:

d(A,B):= | ln(A,B;P,Q) |

In this way, projective geometry, the humblest of all, became "the queen of geometry."

Axioms and theorems

But we still have to address the question of the underlying structure in projective geometry, of making its axioms explicit. The German geometers of the 19th century managed to expose it considering the lattice formed by the linear submanifolds (points, lines, planes, etc.). They characterized it as a 3-dimensional lattice (in the case of space) with the following properties:

1. dim(A+B) = dim(A) + dim(B) - dim(A)B)
2. There are 5 points in general position (no plane passes by 4 of them).
3. Valid Theorem of Desargues.

The Spanish contribution

If it is desired that the field of coordinates be commutative, the validity of Pappus's hexagon theorem must be imposed. In fact, the high school professor Ventura de los Reyes Prósper (Castuera, May 31, 1863 - Toledo, November 27, 1922) wrote a letter to Pasch explaining how in space Desargues' theorem follows from the other two properties and is therefore superfluous. Pasch, astonished at the simplicity of the argument («... auf denkbar einfachste Art... ») which greatly simplified his recent book, published it in 1888 in the Matematische Annalen. It is the first Spanish contribution published in a journal of such importance. In the case of the projective plane, obviously it has to be required that there be a lattice of dimension with 4 points in general position (no line passes through 3 of them); but in such a case Desargues's theorem is no longer a consequence of the other two properties, but must continue to hold as an additional axiom.

In the middle of the 20th century

But the simple projective line resists being characterized as a lattice, since its order relation is absolutely trivial. It was not until the middle of the XX century that a conceptual framework was achieved that encompassed both the structure of the projective line and that of the plane and the projective space. The concept of "scheme" introduced by Grothendieck makes it possible to collect them within him and, as a superabundant gift, also all commutative algebra and a large part of arithmetic.