Reider's theorem

Reider's theorem 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 {displaystyle DE=0,E^{2}=-1} , or {displaystyle 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: {displaystyle DE=0,E^{2}=-1} or {displaystyle -2} ; {displaystyle DE=1,E^{2}=0} or {displaystyle -1} ; {displaystyle DE=2,E^{2}=0} ; {displaystyle 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 ampleness of L implies D · E = m(L · E) ≥ m > 2.

Thus by the first part of Reider's theorem |KX+mL| is base-point-free. Similarly, for any m > 3 the linear system |KX+mL| is very ample.

References Reider, Igor (1988), "Vector bundles of rank 2 and linear systems on algebraic surfaces", Annals of Mathematics, Second Series, Annals of Mathematics, 127 (2): 309–316, doi:10.2307/2007055, ISSN 0003-486X, JSTOR 2007055, MR 0932299 This algebraic geometry–related article is a stub. You can help Wikipedia by expanding it.

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