# ampleness

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In [Kn3] we introduced the notion of birational k-very ampleness: A line bundle [Laplace] is said to be birationally k-very ample if there exists a Zariskiopen, dense subset U of X such that the restriction map [H.sup.0] ([Laplace]) [right arrow] [H.sup.0] ([Laplace] [cross product] [O.sub.z]) is surjective for any 0-dimensional subscheme Z of X of length [h.sup.0] (O.sub.z]) = k + 1 with Supp(Z) C U.
Note that by the ampleness assumption on L we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (S) is smooth if and only if [phi](L) [greater than or equal to] 3, and if [phi](L) = 2, then Sing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] cf.
Many results are however true without the ampleness assumptions: Lemmas 2.3, 2.4(a)-(c), 3.1 and 3.2 are stated in general.
The ampleness assumption first enters the picture in a crucial way in the proof of Proposition 3.3.
so that the only possibility is (l, [L.sup.2], [phi](L), R.L) = (3, 10, 2, 1) by the ampleness of L.
But this contradicts the ampleness of L, as (N - M).
The implications (iii) [??] (ii) [??] (i) [??] (iv) in Theorem 1.1 hold, even without the ampleness assumption on L and assuming only [L.sup.2] > 0.
As R.E > 0 by the ampleness of L, we get that both N and M are nef.
The ampleness of L implies 0 < N.L = 3 + [N.sup.2], whence [N.sup.2] [greater than or equal to] -2.
(N + [DELTA]) < 0, whence R.N [greater than or equal to] 2 by the ampleness of L, so that R.[DELTA] [less than or equal to] -3.
The Kleiman criterion of ampleness implies then that the multiple point Seshadri constants are subject to the following upper bound which depends only on the degree of L and the number of points

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