The purpose of this thesis is to expose some results relative to the asymptotic behaviour of the graded Betti numbers of algebraic varieties, as the positivity of the embedding grows.
The first part presents the basic definitions and results relative to to the language of graded Betti numbers and Koszul cohomology.
The second part is dedicated to the recent results of L. Ein and R. Lazarsfeld relative to the asymtpotic shape of the Betti table of a positive embedding of a smooth variety, and to the conjecture of L. Ein, D. Erman and R. Lazarsfeld of the asymptotic normality of the Betti numbers.
The third part specializes the problem to that of Betti numbers of Veronese embeddings for the projective plane, and presents a technique for computing the graded Betti numbers using representation theory.
The appendix recalls some results on algebraic groups, in particular representations of SL(n)
