“Reider's theorem”的意思、由来-开放百科全书

词条

Reider's theorem

释义

Statement
Applications
References

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 K_X the canonical divisor of X.

If D² > 4, then the linear system |K_X+D| has no base points unless there exists a nonzero effective divisor E such that
- , or
- ;
If D² > 8, then the linear system |K_X+D| is very ample unless there exists a nonzero effective divisor E satisfying one of the following:
- or ;
- or ;
- ;

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 |K_X+mL| is base-point-free. Similarly, for any m > 3 the linear system |K_X+mL| is very ample.

References

{{Citation | doi=10.2307/2007055 | last1=Reider | first1=Igor | title=Vector bundles of rank 2 and linear systems on algebraic surfaces | jstor=2007055 | mr=932299 | year=1988 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=127 | issue=2 | pages=309–316 | publisher=Annals of Mathematics}}

2 : Algebraic surfaces|Theorems in algebraic geometry

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