In the algebraice geometry, one says about "geometrically irreducible closed curve" over field $k$. For example, the theorem 5.4.5 (pp. 147) of ''Heights in Diophantine Geometry'' of E. Bombieri wrote that "Let $C$ be a geometrically irreducible closed curve in $\mathbb{G}^n_m$ defined over a number field $K$, not a translate of a subtorus of $\mathbb{G}^n_m$, and let $\Gamma$ be any finitely generated subgroup of $\mathbb{G}^n_m(\bar{K})$. Then $C \cap \Gamma$ is an effectively computable finite set."
I think it means that a curve $(C)$ which is geometrically irreducible and closed over $k$. This means that
If $(C)$ is represented by $F(x,y) = 0$ with $F(x,y) \in k[X, Y]$ then $F(x,y)$ is irreducible over $\bar{k}[x,y]$ where $\bar{k}$ is the algebraic closure of $k$.
In the graph of $(C)$, the beginning and end points are the same.
My question: Is my definition true? Are there other definition of this problem?
Thank you very much for your interests!