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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!

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    $\begingroup$ 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. $\endgroup$ – Alex Youcis May 6 at 19:25
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    $\begingroup$ 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". $\endgroup$ – yamete kudasai May 6 at 19:44
  • $\begingroup$ @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." $\endgroup$ – mathJuan May 7 at 17:00
  • $\begingroup$ @yametekudasai Thank you so much for your comments! $\endgroup$ – mathJuan May 7 at 17:06

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