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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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Effectiveness of the distinguished theta characteristic in characteristic 2
Let $F:X\to X$ be the relative Frobenius morphism. It is a finite morphism of degree $2$, and we have an exact sequence
$$0\to \mathcal{O}_X\to F_*\mathcal{O}_X \to L \to 0$$
for some invertible shea …