Reider's theorem

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
• $DE\; =\; 0,\; E^2\; =\; -1$, or
• $DE\; =\; 1,\; E^2\; =0$;
• 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:
• $DE\; =\; 0,\; E^2\; =\; -1$ or $-2$;
• $DE\; =\; 1,\; E^2\; =\; 0$ or $-1$;
• $DE\; =\; 2,\; E^2\; =\; 0$;
• $DE\; =\; 3,\; D\; =\; 3E,\; E^2\; =\; 1$

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.

• {{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}}

Category:Algebraic surfaces

Category:Theorems in algebraic geometry

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