Des Dessins

Now, Grothendieck was interested in the Galois group of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$, which acts on the coefficients of the equation of an algebraic curve over $\overline{\mathbb{Q}}$, giving another curve of the same type. Considering the dessins associated to those curves, that group then transforms a dessin into another dessin, and so, dessins can be used to obtain a geometric interpretation of the absolute Galois group.

In Leila Schneps - Dessins d'enfants on the Riemann Sphere you can find a definition of dessins and the Grothendieck correspondence between Belyi morphisms and dessins. It also has pictures of how the cartographic group acts on the flag set of a dessin.

Grothendieck correspondence means that there is a bijection between isomorphism classes of dessins and isomorphism classes of Belyi morphisms (morphisms f:X->P1C which are ramified only over three points).

Des Dessins

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Grothendieck's dessins d'enfants are closely connected to the study of coverings of the threetimes punctured sphere, and such coverings can be considered from many different points of view.In this survey it is shown how all of them are equivalent, and how the absoluteGalois group acts on these objects.

In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings".

Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by Belyi's theorem, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic numbers. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.

Recognizable modern dessins d'enfants and Belyi functions were used by Felix Klein.[2] Klein called these diagrams Linienzge (German, plural of Linienzug "line-track", also used as a term for polygon); he used a white circle for the preimage of 0 and a '+' for the preimage of 1, rather than a black circle for 0 and white circle for 1 as in modern notation.[3] He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with monodromy group P S L ( 2 , 11 ) {\displaystyle PSL(2,11)} , following earlier constructions of a 7-fold cover with monodromy P S L ( 2 , 7 ) {\displaystyle PSL(2,7)} connected to the Klein quartic.[4] These were all related to his investigations of the geometry of the quintic equation and the group A 5 = P S L ( 2 , 5 ) {\displaystyle A_{5}=PSL(2,5)} , collected in his famous 1884/88 Lectures on the Icosahedron. The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of trinity.

Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in 1984 in his Esquisse d'un Programme.[5] Zapponi (2003) quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants:

A vertex in a dessin has a graph-theoretic degree, the number of incident edges, that equals its degree as a critical point of the Belyi function. In the example above, all white points have degree two; dessins with the property that each white point has two edges are known as clean, and their corresponding Belyi functions are called pure. When this happens, one can describe the dessin by a simpler embedded graph, one that has only the black points as its vertices and that has an edge for each white point with endpoints at the white point's two black neighbors. For instance, the dessin shown in the figure could be drawn more simply in this way as a pair of black points with an edge between them and a self-loop on one of the points.It is common to draw only the black points of a clean dessin and to leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map.

Due to Belyi's theorem, the action of {\displaystyle \Gamma } on dessins is faithful (that is, every two elements of {\displaystyle \Gamma } define different permutations on the set of dessins),[12] so the study of dessins d'enfants can tell us much about {\displaystyle \Gamma } itself. In this light, it is of great interest to understand which dessins may be transformed into each other by the action of {\displaystyle \Gamma } and which may not. For instance, one may observe that the two trees shown have the same degree sequences for their black nodes and white nodes: both have a black node with degree three, two black nodes with degree two, two white nodes with degree two, and three white nodes with degree one. This equality is not a coincidence: whenever {\displaystyle \Gamma } transforms one dessin into another, both will have the same degree sequence. The degree sequence is one known invariant of the Galois action, but not the only invariant.

The stabilizer of a dessin is the subgroup of {\displaystyle \Gamma } consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of {\displaystyle \Gamma } and algebraic number fields, the stabilizer corresponds to a field, the field of moduli of the dessin. An orbit of a dessin is the set of all other dessins into which it may be transformed; due to the degree invariant, orbits are necessarily finite and stabilizers are of finite index. One may similarly define the stabilizer of an orbit (the subgroup that fixes all elements of the orbit) and the corresponding field of moduli of the orbit, another invariant of the dessin. The stabilizer of the orbit is the maximal normal subgroup of {\displaystyle \Gamma } contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of Q {\displaystyle \mathbb {Q} } that contains the field of moduli of the dessin. For instance, for the two conjugate dessins considered in this section, the field of moduli of the orbit is Q ( 21 ) {\displaystyle \mathbb {Q} ({\sqrt {21}})} . The two Belyi functions f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.[13]

(14) L'apprciation du caractre individuel d'un dessin ou modle devrait consister dterminer s'il existe une diffrence claire entre l'impression globale qu'il produit sur un utilisateur averti qui le regarde et celle produite sur lui par le patrimoine des dessins ou modles, compte tenu de la nature du produit auquel le dessin ou modle s'applique ou dans lequel celui-ci est incorpor et, notamment, du secteur industriel dont il relve et du degr de libert du crateur dans l'laboration du dessin ou modle.

(21) La nature exclusive du droit confr par le dessin ou modle communautaire enregistr correspond la volont de lui donner une scurit juridique plus grande. En revanche, le dessin ou modle communautaire non enregistr ne devrait confrer que le droit d'empcher la copie. La protection ne peut donc s'tendre des produits auxquels sont appliqus des dessins ou modles qui sont le rsultat d'un dessin ou modle conu de manire indpendante par un deuxime crateur. Ce droit devrait galement tre tendu au commerce des produits auxquels sont appliqus des dessins ou modles dlictueux.

1. Un dessin ou modle communautaire enregistr est dclar nul sur demande introduite auprs de l'Office, conformment la procdure prvue aux titres VI et VII, ou par un tribunal des dessins ou modles communautaires la suite d'une demande reconventionnelle dans le cadre d'une action en contrefaon.

3. Un dessin ou modle communautaire non enregistr est dclar nul par un tribunal des dessins ou modles communautaires sur demande introduite auprs dudit tribunal ou la suite d'une demande reconventionnelle dans le cadre d'une action en contrefaon.

2. Lorsque la demande est dpose auprs du service central de la proprit industrielle d'un tat membre ou auprs du Bureau Benelux des dessins ou modles, ce service ou ce Bureau prend toutes les mesures ncessaires pour transmettre la demande l'Office dans un dlai de deux semaines aprs son dpt. Il peut exiger du demandeur une taxe qui ne dpasse pas le cot administratif correspondant la rception et la transmission de la demande.

1. Plusieurs dessins et modles peuvent tre combins en une demande d'enregistrement multiple de dessins ou modles communautaires. Sauf lorsqu'il s'agit d'ornementations, cette possibilit est subordonne la condition que les produits dans lesquels les dessins ou modles sont destins tre incorpors ou auxquels ils sont destins tre appliqus fassent tous partie de la mme classe de la classification internationale pour les dessins et modles industriels.

4. Chacun des dessins ou modles compris dans une demande multiple ou un enregistrement multiple peut tre trait indpendamment des autres aux fins du prsent rglement. Il peut notamment, indpendamment des autres, tre mis en oeuvre, faire l'objet de licences, de droits rels, d'une excution force, tre compris dans une procdure d'insolvabilit, faire l'objet d'une renonciation, d'un renouvellement, d'une cession, d'un ajournement de la publication ou tre dclar nul. Une demande multiple ou un enregistrement multiple ne peut tre divis en demandes indpendantes ou en enregistrements indpendants que dans les conditions prvues par le rglement d'excution. 2351a5e196