I have a simple question for you (I think ?).
We take a point $a \in \mathbb{P}^n(\mathbb{K})$. I know that $I(a) (X_0,...,X_n) $ since $P \in I(a)$ must be homogenous. So $I(a)$ is not a maximal ideal. But can we calculate explicitly $I(a)$ ?
Can we found polynomials such that $V(P_1,...,P_r)=\{a\} ?$ In the affine space, it's very easy to see that $P_1=X_1-a_1, ..., P_n=X_n-a_n$ works but in the projective space, it doesn't work...
Cordially, Doeup
$$I(a)=\langle a_iX_j-a_jX_i\vert i,j=0,\cdots,n\rangle$$