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 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
- $D E = 0, E^{2} = - 1$ , or
- $D E = 1, E^{2} = 0$ ;
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:
- $D E = 0, E^{2} = - 1$ or $- 2$ ;
- $D E = 1, E^{2} = 0$ or $- 1$ ;
- $D E = 2, E^{2} = 0$ ;
- $D E = 3, D = 3 E, E^{2} = 1$

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.

References

Reider, Igor (1988), "Vector bundles of rank 2 and linear systems on algebraic surfaces", Annals of Mathematics, Second Series, 127 (2), Annals of Mathematics: 309–316, doi:10.2307/2007055, ISSN 0003-486X, JSTOR 2007055, MR 0932299

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