This article deals with practical conditions for tuning a subdivision scheme in order to control its artifacts in the vicinity of a mark point. To do so, we look for good behaviour of the limit vertices rather than a good mathematical property of the limit surface. The good behaviour of the limit vertices is characterized with the definition of C 2 -convergence of a scheme. We propose necessary explicit conditions for C 2 -convergence of a scheme in the vicinity of any mark point being a vertex of valency greater or equal to three.