On my notes I have the following statement:
Let $F_{1},\dots,F_{r}$ be homogeneous polynomials in $k[x_{0},\dots,x_{N}]$ ($k$ algebraically closed field) with $r\leq N$. Then $X=V(F_{1},\dots,F_{r})\subseteq\mathbb{P}^{N}$ is non empty and every irreducible component of $X$ has dimensione at least $N-r$.
The proof for the non-emptyness is done by induction on $r$. But don't we need some extra hypothesis to guarantee that the locus is non-empty? I.E: let's start with $r=1$. Then $X=V(F_{1})$ is a hypersurface if the polynomial is not constant (and homogeneous). How do we know that from the hypothesis?
Thanks for helping !