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For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.
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On morphisms to projective space arising from a linear system
Now, divisors on $X$ give rise to divisors on $X_0$ by restriction, so we can reduce our problem to showing that $C$ is either contained in or disjoint from any divisor of $X_0$ that is linearly equivalent …