Grothendieck'in desen teorisi

The bridge between algebraic geometry and complex geometry is built by Riemann on the following observation: compact Riemann surfaces and nonsingular complex projective curves can be considered to be same. After the celebrated theorem of Belyi, which is a bridge between curves defined over number fields and the existence of certain coverings of the projective line, Grothendieck launched in the 1980s, in his famous Equisse d'un programme that such coverings is completely determined by the preimage of the real interval [0,1] which is named a (child's drawing) by him. Belyi showed that every algebraic curve defined over algebraic closure of Q can be represented as a covering of the projective line ramified at most three points. In other words, every algebraic curve defined over algebraic closure Q contains an embedded dessin d'enfant. We give an introduction to the theory of dessins d'enfants. A dessin can be regarded as an ordered pair of permutations generating a transitive subgroup of a symmetric group on n letters. The group $\pgl$ has an action on these pairs of permutation, hence on dessins d'enfants. Our aim is to define and study an action of $\pgl$ on dessins which appears to have not been studied until now. The final section is dedicated to investigate combinatorial and arithmetic aspects of this action.