Ample Line Bundles on Blown Up Surfaces

1994/10/12 by Oliver Küchle, Küchle, Oliver
Engineering · Mathematics · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Physics and Engineering Research Articles #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9410011

4 pages, AMS-TeX 2.1

arxiv created 1994/10/12 · openalex publication_date 1994/10/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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,..., E_k be the exceptional divisors. We show that L = nH' -E₁ - ... -E_k is ample if and only if L² is positive provided the integer n is at least 3.

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