Let $V$ be a projective variety defined over $\mathbb{F}_p$. For any extension of $\mathbb{F}_p$ let $V(k)$ be the $k$-points of $V$, that is, if $V$ is defined by the homogeneous polynomials $f_1,\ldots,f_r\in\mathbb{F}_p[X_0,\ldots,X_n]$, then $V(k)=\{(x_0:\ldots :x_n)\in k^n:f_i(x_0,\ldots,x_n)=0\ \forall i\}$. Then we can talk about dimension of $V(k)$ for any $k$ because dimension is just a concept of topological spaces and all $V(k)$ are topological spaces with the Zariski topology.
1-Are all the numbers $\dim V(k)$ equal? (assume $k$ runs over subfields of $\bar{\mathbb{F}_p}$). If not, when we read the expression "Let $V$ be a projective variety defined over $\mathbb{F}_p$ of dimension $r$", do we have to read $r=\dim V(\bar{\mathbb{F}_p})$?
2-Similarly I have read the expression "Let $V$ be a smooth projective variety defined over $\mathbb{F}_p$". The concept of smoothness I know is in term of dimension, that is, $V$ is smooth if the rank of the matrix $[\partial f_i/\partial x_j]$ in each $(a_0:\ldots:a_n)\in V$ is $n-\dim V$. So my question is which is the definition of that expression?.
