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 [line bundle|nef] divisor on a smooth projective surface X. Denote by K_X the canonical divisor of X.

If D² > 4, then the linear [system of divisors|linear system] |K_X+D| has no base points unless there exists a nonzero effective divisor E such that
*, or
* ;
If D² > 8, then the linear system |K_X+D| is very ample unless there exists a nonzero effective divisor E satisfying one of the following:
* or ;
* or ;
* ;
*

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 D''² = 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 |K_X+mL| is base-point-free. Similarly, for any m > 3 the linear system |K_X+mL| is very ample.