Reider's theorem

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

Statement[edit]

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.

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