# Intersection of a hyperplane and a curve

Let us fix a projective curve X over a field k. With nt, I mean a variety with all irreducible components of dimension 1. Let us suppose that there is a smooth rational point $x \in X$. My question is:

Is it possible to find a hyperplane such that the intersection of X and this hyperplane is a point? If so, how?

If you are looking for some projective immersion for your curve such that this happens, then if your curve is irreducible just consider the divisor $(2g+1)p$, where $p$ is the point you were talking about and $g$ is the genus of the curve. This divisor is very ample, and so if we immerse the curve in projective space with this divisor, there exists a hyperplane such that the set theoretical intersection of the hyperplane with the curve is just $p$.