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.
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 ampleness of L implies D · E = m ≥ m > 2.
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.
