We begin this chapter with a discussion of various models that can be attached to a projective variety and explore in much more detail the leads developed in Chapter 1. One idea was to attach to a projective variety X a preferred member X min of its birational equivalence class with good properties, such as K X min nef. It is then minimal in the sense that any bi-rational map from X min to a smooth projective variety is an isomorphism (Proposition 1.45). This is possible only if X is not uniruled and the (explicit) construction of X min is one of the purposes of Mori’s minimal model program.
KeywordsProjective Variety Exceptional Divisor Effective Divisor Smooth Projective Variety Cartier Divisor
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