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
Applications Reider's theorem implies the surface case of the Fujita conjecture. Let L be an ample line bundle on a smooth projective surface X. If m > 2, then for D=mL we have
• D2 = m2 L2 ≥ m2 > 4;
• for any effective divisor E the ampl
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