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Example text

Slightly later, Angehrn and Siu [AS95], [Siu95] introduced other closely related methods, producing better bounds for the global generation question; since their method is rather delicate, we can only refer the reader to the above references. In the sequel, X denotes a projective algebraic n-dimensional manifold. The first observation is the following well-known consequence of the Riemann-Roch formula. 1) Special case of Riemann-Roch. Let J ⊂ OX be a coherent ideal sheaf on X such that the subscheme Y = V (J ) has dimension d (with possibly some lower dimensional components).

2 a), the condition P (m) ≥ 1 is achieved for some m ∈ [bk , bk + n] and we can take η = n. , the algebraic metric defined with all sections of µL vanishing at x0 ). Hence Fa′ = aKX + (a(n + 1) + 1)L + nµL has a metric h′a such that V (I(h′a )) is zero dimensional and contains {x0 }. 12), we conclude that KX + Fa′ = (a + 1)KX + (a(n + 1) + 1 + nµ)L is generated by sections, in particular KX + a(n+1)+1+nµ L is nef. As a tends to +∞, a+1 we infer that KX + (n + 1)L is nef. 4 a). Here, the choice a = 1 is sufficient for our purposes.

Fix m > 0 so that Φ = Φ|mL| has generic rank r. Select a nonsingular ample divisor A in X such that the restriction of Φ|mL| to A still has rank r (for this, just take A passing through a point x ∈ / B|mL| at which rank(dΦx ) = r < n, in such a way that the tangent linear map dΦx↾TA,x still has rank r). Then κ(L↾A ) ≥ r = κ(L) (we just have an equality because there might exist sections in H 0 (A, mL↾A ) which do not extend to X). On the other hand, we claim that ν(L↾A ) = k = ν(L). The inequality ν(L↾A ) ≥ ν(L) is i clear.

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(83, 64)-Konfigurationen in Laguerre-, Mobius-und weiteren Geometrien by Benz W.

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