When dealing with a Support Vector Machine (SVM) with a strictly positive definite kernel, a common misconception is that the main handle for controlling the nonlinearity of the classification surface is the set of kernel hyperparameters. We show here that this is not the case: in particular, we prove that, regardless of the value of the kernel hyperparameter, it is always possible to tune the nonlinearity of the classifier by acting only on the regularization hyperparameter C, even achieving perfect learning of any non-degenerate training set.