Connectedness and Bertini Theorems
This chapter concentrates on the connectedness and Bertini theorems that will be needed in Chapters 4 and 5 to prove a result on the existence of components of the Stückrad-Vogel cycle in all possible dimensions and a converse to Bezout’s Theorem. Section 3.1 introduces the suitably inductive type of connectedness, namely connectedness ‘in a given dimension’, that we need and establishes Grothendieck’s Connectedness Theorem which shows that this type of connectedness behaves well under the taking of hyperplane (indeed hypersurface) sections. Typically, this theorem is deduced from a local version via coning, the local version being given an algebraic proof. Once again, stability under deformation to the normal cone is considered.
KeywordsIrreducible Component Local Ring Open Dense Subset Cartier Divisor Closed Subscheme
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