By Carides G. W.

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**Extra info for A regression-based method for estimating mean treatment cost in the presence of right-censoring (200**

**Example text**

8,a) that n = κ(LN (−D)) ≤ κ(LN ) = κ(L). 52 H. Esnault, E. 4,d). Hence we ﬁnd a divisor C > 0 and µ > 0 with Lµ (−C) ample. 1) H b (X, L−1 ⊗ (L−N ·ν−µ (ν·D + C))η ) = 0 for b < n and for η suﬃciently large. 13) this condition is compatible with blowing ups. Hence we may assume D + C to be a normal crossing divisor and, choosing ν large enough, we may again assume that the multiplicities of D = ν·D + C are smaller than N = N ·ν + µ. Replacing D and N by some high multiple we can assume in addition that LN (−D ) is generated by global section and that H b (X, L−N −1 (D )) = 0 for b < n.

Lemma. For an invertible sheaf L on a projective manifold X the following two conditions are equivalent: a) L is numerically eﬀective. b) For an ample sheaf A and all ν > 0 the sheaf Lν ⊗ A is ample. Proof: By Seshadri’s criterion A is ample if and only if for some all curves C in X deg (A |C ) ≥ · m(C) > 0 and where m(C) is the maximal multiplicity of points on C. 8. Lemma. 11) and that dim X ≤ p). Then one has: §5 Vanishing theorems for invertible sheaves 47 a) κ(L) = n = dim X, if and only if c1 (L)n > 0 (where c1 (L) is the Chern class of L).

Moreover, ResDj (∇) ∈ ZZ for j = 1, . . , r. For a component Aj of A we have ResAj (∇) = 0, and for a component Bj of B we have ResBj (∇) = 1. 13). 3. Corollary. 1), assume that LN = OX (D) for a normal r crossing divisor D = j=1 αj Di with 0 < αj < N and assume that there exists an ample eﬀective divisor B with Bred = Dred . Then H b (X, L−1 ) = 0 for b < n. 7,c). 12,d) for κ(LN (−D)) = dim X. Moreover, we may blow up, whenever we like. We can write (replacing N and D by some multiple) LN (−D) = A(Γ) for some eﬀective divisor Γ and some ample sheaf A.

### A regression-based method for estimating mean treatment cost in the presence of right-censoring (200 by Carides G. W.

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