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Local Cohomology, Second Edition by R. Y. Sharp, M. P. Brodmann

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19

Connectivity in algebraic varieties

The study of the topological connectivity of algebraic sets is a fundamental subject in algebraic geometry. Local cohomology is a powerful tool in this field. In this chapter we shall use this tool to prove some results on connectivity which are of basic significance. Our main result will be the Connectedness Bound for Complete Local Rings, a refinement of Grothendieck’s Connectedness Theorem. We shall apply this result to projective varieties in order to obtain a refined version of the Bertini–Gothendieck Connectivity Theorem. Another central result of this chapter will be the Intersection Inequality for Connectedness Dimensions of Affine Algebraic Cones. As an application it will furnish a refined version ...

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