In the present paper we consider the case of a general $\cont{r+2}$ perturbation, for $r$ large enough, of an a priori unstable Hamiltonian system of $2+1/2$ degrees of freedom, and we provide explicit conditions on it, which turn out to be $\cont{2}$ generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion.

This is a generalization of the result in Delshams et al., \emph{Mem. Amer. Math. Soc.}, 2006, where it was considered the case of a perturbation with a finite number of harmonics in the angular variables.

The method of proof is based on a careful analysis of the geography of resonances created by a generic perturbation and it contains a deep quantitative description of the invariant objects generated by the resonances therein. The scattering map is used as an essential tool to construct transition chains of objects of different topology. The combination of quantitative expressions for both the geography of resonances and the scattering map provides, in a natural way, explicit computable conditions for instability.