I'm hoping to see how the following bound is reached.
For an algebraic variety $V\subset\mathbb{A}^n$ over some field $F$, one defines $\dim V=\operatorname{trdeg}(F(x)/F)$ for a generic point $(x)$ of $V$. Also, I denote by $V(f_1,\dots,f_m)$ the set of zeroes in $\mathbb{A}^n$ of some homogeneous forms $f_i$.
All right, so $V:=V(f_1,\dots,f_m)$ is a homogeneous algebraic variety. How does it follow that the dimension of $V$ is bounded below by $n-m$? That is, why is $\dim V\geq n-m$? Thank you.