Reider's theorem

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 Thus by the first part of Reider's theorem |KX+mL| is base-point-free. Similarly, for any m > 3 the linear system |KX+mL| is very ample.
 * D2 = m2 L2 ≥ m2 > 4;
 * for any effective divisor E the ampleness of L implies D · E = m(L · E) ≥ m > 2.