The aim of this chapter is to discuss some auxiliary results that do not fit anywhere else. Section 1 considers rational curves on exceptional loci of maps. The main result (1.2) is a theorem of [Abhyankar56] which says that if f : X → Y is a birational morphism and Y is smooth, then every exceptional divisor of f is ruled. This is clear if resolution of indeterminacies holds for f -1, but the result can be proved using only some partial resolution results. The relevant assertion about resolutions (1.3) is contained in [Zariski39] and it is useful in many instances.
KeywordsLine Bundle Exceptional Divisor Rational Curf Coherent Sheaf Cartier Divisor
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