Concept

Reider's theorem

Summary
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 ** DE = 0, E^2 = -1, or ** 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: ** DE = 0, E^2 = -1 or -2; ** DE = 1, E^2 = 0 or -1; ** DE = 2, E^2 = 0; ** 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 ampl
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications

No results

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading