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Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the compactification of $U$ to $V$?

EDIT: the question considers the smooth compactification of $U$.

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the compactification of $U$ to $V$? Note that we are not merely asking that the map $U\to V$ does not extend.

EDIT: the question considers the smooth compactification of $U$.

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the smooth compactification of $U$ to $V$?

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the compactification of $U$ to $V$?

EDIT: the question considers the smooth compactification of $U$.

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the compactification of $U$ to $V$?

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the smooth compactification of $U$ to $V$?