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I'm not using the language of schemes. I define projective varieties to be an algebraic set in $\mathbb P^n$ with the homogeneous ideal to be a prime ideal, and a projective curves to be $1$-dimension projective varieties.

Is there any proof of any morphisms between curves being either constant or surjective that does not use the language of schemes?

OrthoPole
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  • Does https://math.stackexchange.com/questions/1747174/a-morphism-from-a-projective-curve-x-to-a-curve-y-is-either-constant-or-surj help ? Even if I dislike the book, you can find some stuff in Shafarevich's book "Basic Algebraic Geometry I". – NaNoS Mar 08 '24 at 09:00
  • You should assume that curves (or varieties) are irreducible, otherwise it is not true (e.g. $C \hookrightarrow C \cup D$). – red_trumpet Mar 08 '24 at 09:41
  • @red_trumpet Yeah. I assume the ideal of varieties to be prime. – OrthoPole Mar 09 '24 at 11:21

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