Reider's theorem

In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.

Let D be a nef divisor on a smooth projective surface X. Denote by KX the canonical divisor of X. 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.