The main topic of this chapter is a detailed study of flat morphisms. Although it is impossible to fully capture the algebraic notion of flatness in geometric terms, there is a useful heuristic: If f : X → S is a flat morphisms, then the fibers f−1(s) form a continuously varying family of varieties, as s varies in S. In simple situations, the converse to this statement is true, see Proposition 14.14, Theorem 14.32 and Theorem 14.126 below. One could say that flatness is the correct generalization of this naive point of view to the general case.
KeywordsPrime Ideal Irreducible Component Local Ring Galois Extension Hilbert Scheme
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