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Concept# Reider's theorem

Summary

In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.
Statement
Let D be a nef divisor on a smooth projective surface X. Denote by KX the canonical divisor of X.

- If D2 > 4, then the linear system |KX+D| has no base points unless there exists a nonzero effective divisor E such that ** DE = 0, E^2 = -1, or ** DE = 1, E^2 =0 ;
- If D2 > 8, then the linear system |KX+D| is very ample unless there exists a nonzero effective divisor E satisfying one of the following: ** DE = 0, E^2 = -1 or -2; ** DE = 1, E^2 = 0 or -1; ** DE = 2, E^2 = 0; ** DE = 3, D = 3E, E^2 = 1

- D2 = m2 L2 ≥ m2 > 4;
- for any effective divisor E the ampl

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