Reider's theorem
Encyclopedia
In algebraic geometry
, Reider's theorem gives conditions for a line bundle
on a projective surface to be very ample.
, 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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, Reider's theorem gives conditions for a line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
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 nefNef
-Historical:* Nef an extravagant table-ornament in the shape of a ship, from* a French name for a type of ship called a carrack in English* the French word for nave of a church-People:* Alain Nef , Swiss footballer...
, 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.