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.

  • If D2 > 4, then the linear system |KX+D| has no base points unless there exists a nonzero effective divisor E such that D E = 0 , E 2 = − 1 {\displaystyle DE=0,E^{2}=-1}, or D E = 1 , E 2 = 0 {\displaystyle DE=1,E^{2}=0};
  • If D2 > 8, then the linear system |KX+D| is very ample unless there exists a nonzero effective divisor E satisfying one of the following: D E = 0 , E 2 = − 1 {\displaystyle DE=0,E^{2}=-1} or − 2 {\displaystyle -2}; D E = 1 , E 2 = 0 {\displaystyle DE=1,E^{2}=0} or − 1 {\displaystyle -1}; D E = 2 , E 2 = 0 {\displaystyle DE=2,E^{2}=0}; D E = 3 , D = 3 E , E 2 = 1 {\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 |KX+mL| is base-point-free. Similarly, for any m > 3 the linear system |KX+mL| is very ample.

  • 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:, ISSN, JSTOR, MR