The Stacks project

Exercise 111.31.3. Let $X \subset \mathbf{C}^ n$ be an affine variety. Let us say $X$ is a cone if $x = (a_1, \ldots , a_ n) \in X$ and $\lambda \in \mathbf{C}$ implies $(\lambda a_1, \ldots , \lambda a_ n) \in X$. Of course, if $\mathfrak p \subset \mathbf{C}[x_1, \ldots , x_ n]$ is a prime ideal generated by homogeneous polynomials in $x_1, \ldots , x_ n$, then the affine variety $X = V(\mathfrak p) \subset \mathbf{C}^ n$ is a cone. Show that conversely the prime ideal $\mathfrak p \subset \mathbf{C}[x_1, \ldots , x_ n]$ corresponding to a cone can be generated by homogeneous polynomials in $x_1, \ldots , x_ n$.

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