Suppose I have homogeneous polynomials $f_1, .., f_r \in \mathbb{C}[x_1, ..., x_n]$, and let $I = (f_1, ..., f_r)$. Let $V:=V(I) \subseteq \mathbb{C}^n$ be the points where $f_i$'s vanish. Suppose $V$ can be expressed as $$ V = V_1 \cup V_2, $$ where $V_1$ and $V_2$ are non-empty Zariski closed sets. Let $V_i = V(I_i)$. Does it then follow that $I_i$ is an ideal generated by homogeneous polynomials $(i=1,2)$ as well?
I am asking this because I was wondering if every irreducible component of $V$ must contain the $\mathbf{0} = (0, ..., 0)$ point or not.
Thank you very much!
PS I would like to add why I asked this question here. I was learning about codimension of intersection of varieties. In Basic question related to dimension of intersection of two varities, I was pointed out that when $V$ and $W$ are not irreducible algebraic sets then the formula $$ codim \ V + codim \ W \geq codim (V \cap W) $$ does not necessarily hold (But it holds true when $V$ and $W$ are affine irreducible varieties that have non-trivial intersection). What I was wondering was that perhaps this inequality holds true even when $V$ and $W$ are not necessarily irreducible in the projective setting. I wanted to prove it, so I wanted to think about the zero sets of homogeneous polynomials in the setting $\mathbb{C}^n$ so that I could apply the inequality where I know it holds true.