Reider's theorem explained

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 K_X the canonical divisor of X.

• If D² > 4, then the linear system |K_X+D| has no base points unless there exists a nonzero effective divisor E such that

DE = 0 , E ^{2} = -1 , orDE = 1 , E ^{2} =0 ;

• 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:

DE = 0 , E ^{2} = -1 or -2 ;DE = 1 , E ^{2} = 0 or -1 ;DE = 2 , E ^{2} = 0 ;DE = 3 , D = 3 E, 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 ampleness of L implies D · E = m(L · E) ≥ 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.