Projective Nullstellensatz

Let $I\subseteq k[x_0,\dots ,x_n]$ be a homogeneous ideal over an algebraically closed field, and letbe the irrelevant ideal. Then $V_{+}(I)=\varnothing$ exactly when either $I=(1)$ or $\sqrt{I}=\mathfrak{m}$. For a proper homogeneous ideal this is also equivalent to ${\mathfrak{m}}^N\subseteq I$ for some $N$.

In the standard graded homogeneous coordinate ring $S=k[x_0,\dots ,x_n]$ of ${\mathbb{P}}^n$, the irrelevant ideal isIt vanishes only at the origin of the corresponding affine cone, which does not represent a projective point.