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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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"rationality" of divisors
What happens with divisors? If $D$ is a divisor on $X$, with irreducible components $D_j$, is it possible to associate a finite extension of $k$ to each $D_j$? …