Ample Line Bundles on Blown Up Surfaces

Abstract

Given a smooth complex projective surface S and an ample divisor H on S, consider the blow up of S along k points in general position. Let H' be the pullback of H and E1,..., Ek be the exceptional divisors. We show that L = nH' -E1 - ... -Ek is ample if and only if L2 is positive provided the integer n is at least 3.

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