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

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, Annals of Mathematics, 127 (2): 309–316, doi:10.2307/2007055, ISSN 0003-486X, JSTOR 2007055, MR 0932299