# What is the definition of ''geometrically irreducible closed curve''?

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!

• Can you tell us what text you are reading? The phrase 'closed' is non-standard and I would assume that it means 'complete' or 'projective' in classical terminology, but I can't be sure. – Alex Youcis May 6 at 19:25
• I think your definition of closed is for $k=\mathbb{R}$ and even there I am not sure how to interpret "beginning point" and "end point". – yamete kudasai May 6 at 19:44
• @AlexYoucis Thank you so much! I am saying about the theorem 5.4.5 in page 147 of ''Heights in Diophantine Geometry'' of E. Bombieri. "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." – mathJuan May 7 at 17:00
• @yametekudasai Thank you so much for your comments! – mathJuan May 7 at 17:06