Reider's theorem

In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.

Statement

Suppose that L is a line bundle on a smooth projective surface with canonical bundle K. Then Reider's theorem states that if L is nef, L² ≥ 10, and two (possibly infinitely near) points x and y are not separated by L + K then there is effective divisor D containing x and y with D . L = 0, D² = −1 or −2, or D.L = 1, D² = 0 or −1, or D.L = 2, D² = 0.

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