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In this PhD thesis we will discuss some aspects in Commutative Algebra whichhave interactions with Algebraic Geometry, Representation Theory andCombinatorics. In particular, in the first chapter we will focus onunderstanding when certain cohomology modules vanish, a classical problemraised by Grothendieck. In the second chapter we will use local cohomology tostudy the connectedness behavior during a Groebner deformation and thearithmetical rank of certain varieties. In the third chapter, we willinvestigate the relations between the minors of a fixed size of a genericmatrix by using tools from the representation theory of the general lineargroup (the results of this chapter will appear in a joint paper with Bruns andConca). In the last chapter we will use combinatorial methods to study theCohen-Macaulay property of the symbolic powers of Stanley-Reisner ideals. Inthe thesis are included five appendixes with some basic needed facts and apreliminary chapter introducing to local cohomology.

Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties