Posts filed under ‘projective geometry’

Experimental algebraic geometry I : the grassmannian

I just began playing with Macaulay 2 to see how it could help doing algebraic geometry without manual tedious computations. Let’s try with the grassmannian: fortunately, the program comes with lots of pre-written functions, including the generation of Grassmanians.

Macaulay provides a command-line interface using the readline library (like many other command-line programs) : here is what input/output looks like

i1 : V = Grassmannian(1,3)
o1 = ideal(p   p    - p   p    + p   p   )
            1,2 0,3    0,2 1,3    0,1 2,3
o1 : Ideal of ZZ[p   , p   , p   , p   , p   , p   ]
                  0,1   0,2   1,2   0,3   1,3   2,3

When I type a command at i1, I get an output o1 with a value \mathrm{ideal}(p_{1,2} p_{0,3} - p_{0,2} p_{1,3} + p_{0,1} p_{2,3}) and a type: this output is an ideal of the ring \mathbb{Z}[p_{0,1}, p_{0,2}, p_{1,2}, p_{0,3}, p_{1,3}, p_{2,3}]. Many features of the Grassmannian as an algebraic variety are available: first define

i1 : V = Grassmannian(1,3,CoefficientRing => QQ);
o1 : Ideal of QQ[p   , p   , p   , p   , p   , p   ]
                  0,1   0,2   1,2   0,3   1,3   2,3
i2 : X = Proj(ring V / V)
o2 = X
o2 : ProjectiveVariety

the projective variety X over \mathbb Q defined by the homogeneous ideal V : here ring V denotes the ambient ring of V. We see that V is a non-singular quadric in 5-dimensional projective space, and check several well-known facts (more…)


7 February 2009 at 11:17 pm Leave a comment

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 {M = [a_1:a_2:a_3:a_4]} and {N = [b_1:b_2:b_3:b_4]}. Then we define the Plücker coordinates of the line MN by the six numbers x_{ij} = (a_i b_j - a_j b_i). 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 (\lambda,\mu) and (\nu,\pi)), giving new coordinates
(\lambda a_i + \mu b_i)(\nu a_j + \pi b_j) - (\lambda a_j + \mu b_j)(\nu a_i + \pi b_i)
= (\lambda \pi - \mu \nu)(a_i b_j - a_j b_i)
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.


5 February 2009 at 11:52 am 1 comment

Les droites de l’espace et la grassmannienne

Le calcul de Schubert désigne un ensemble de techniques destinées à calculer les propriétés énumératives, ou numériques, d’objets communs de l’algèbre linéaire (voir notamment les excellents ouvrages de William Fulton à ce propos). L’exemple traditionnellement choisi, et probablement le plus simple, concerne l’ensemble des droites de l’espace (à 3 dimensions).

Une droite de l’espace est habituellement repérée par sa direction (qui est une droite vectorielle, et dépend donc de deux paramètres, et sa position, qui dépend de deux paramètres supplémentaires (à direction fixée). L’ensemble des droites peut donc être paramétré par quatre paramètres, on peut montrer qu’il est de dimension quatre. Plus intéressant encore, si on place dans l’espace deux plans parallèles, presque toutes les droites peuvent être décrites (de manière unique) par leurs points d’intersection avec ces plans : on obtient ainsi une description par 4 fractions rationnelles (à supposer qu’on sache ce qu’on est en train de paramétrer, ce qui sera plus clair dans une seconde). Ce paramétrage est à peu de chose près bijectif (il manque les droites un peu particulières) : on dit que l’ensemble des droites forme une variété rationnelle.

Le plongement de Plücker

Pour éviter de fastidieuses études de cas, on s’intéresse également aux droites de l’espace projectif : on peut les repérer par les coordonnées de Plücker. Étant donnée une droite de l’espace, considérons deux points sur cette droite de coordonnées projectives {M = [a_1:a_2:a_3:a_4]} et {N = [b_1:b_2:b_3:b_4]}. Les coordonnées de Plücker de la droite sont, par définition, les 6 nombres x_{ij} = (a_i b_j - a_j b_i). Si on avait choisi d’autres points (qui seraient donc des barycentres de M et N), on aurait obtenu des nombres de la forme

(\lambda a_i + \mu b_i)(\nu a_j + \pi b_j) - (\lambda a_j + \mu b_j)(\nu a_i + \pi b_i)
= (\lambda \pi - \mu \nu)(a_i b_j - a_j b_i)

qui sont en fait proportionnels à ceux calculés avec M et N. Les droites sont donc naturellement paramétrées par des coordonnées projectives et forment la variété grassmannienne des droites de l’espace.


30 January 2009 at 8:33 am 1 comment

Inversion and n-circular curves

Inversion is a transformation of the Euclidean plane closely related to circles. It is seldom taught in high schools (at least in France), but it has special interest when dealing with circles, since it maps circles and lines to circles or lines. Inversion is defined with respect to a center O and a radius ρ (in French, the inversion is said to have pôle O and puissance (power) ρ²).


30 December 2008 at 9:08 pm Leave a comment

Circles in projective geometry

Euclidean geometry is closely related to the ability to define circles, which exists naturally in two settings: the complex projective line and the real projective plane.

The Riemann sphere

The complex projective line \mathbb P^1(\mathbb C) is a natural compactification of the Euclidean plane: its natural automorphisms are homographies or Möbius transformations, z \mapsto \frac{az+b}{cz+d} which transform lines or circles into lines or circles. The Euclidean geometry arises from the natural isomorphism between the group of homographies stabilizing the point at infinity and the group of direct isometries of the plane SO_2(\mathbb R^2) \ltimes \mathbb R^2.


30 December 2008 at 12:27 pm 1 comment

Un peu de géométrie plane…

Commençons par un peu de géométrie élémentaire… Soit ABC un triangle non dégénéré dans le plan euclidien. L’usage de coordonnées barycentriques permet de repérer un point M du plan par trois nombres réels {[x:y:z]} de sorte que M soit le barycentre des sommets du triangle avec les poids indiqués.

Let’s begin with elementary geometry… Let ABC be a non-degenerate triangle in the Euclidean plane. By using barycentric coordinates, any point M in the plan can be assigned a triple of real numbers {[x:y:z]}, such that M is the barycentre of the vertices with these weights.


23 December 2008 at 11:47 pm Leave a comment