minimum degree of generators of a vanishing ideal of finite points

Question

Suppose I have finite set of points, for simplicity let us say they are in $\mathbb C^2$ (or the projective space $\mathbb P^2(\mathbb C)$). So let these points be $P_1,P_2,\dots,P_n$. I want to compute the generator of the vanishing ideals of these points, say $I$, and from these I want to magically conclude what the minimum (total/homogenous) degree of a polynomial in $I$ will be. This will not be the minimum degree of the polynomials I get from taking the Gröbner basis will it? Can we say something about this degree if these points are in generic/general position?