I'm writing my thesis and at a certain point I claim, and indeed it turns out to be true from discussions with my supervisors, that certain configurations for n-tuples of points, in a curve of degree k, have the following property:
If there is an n-tuple of points of $\mathbb{P}^2$ such that the configuration C is possible than the configuration C is possible for any n-tuple
For example it's possible that given 8 points on an irreducible quartic one of them is a triple point and the other 7 are non-singular points, but then I claim that given any 8-uple in $\mathbb{P}^2$ there is a quartic with a triple point in one of them and passing through the other 7.
I claim that these configurations are all possible configuration on curves of degree bigger than 3 if we consider an 8-uple.
What I need is a reference for such theorem that I'm quite sure exists
I couldn't find a complete answer. However it turns out that the best way to exclude a configuaration is to use Bezout theorem.
For example a cubic can have at most one singularity, if it had two and we considered a line through the two we woud get an intersection mutliplicity of 2+2=4 that is bigger than the 3*1=3 coming from Bezout.
In some case we need to use singular curves, for example a singular cubic with the signularity (a double point) on the triple point of a quintic has intersection multiplicity 3*2=6 on the point alone