Executive Summary : | If R is a Noetherian ring and I an ideal in R, then the question of minimal pairs of (r,s) such that I^(r)? I^s has been a hot topic of research for the past couple of decades. To study this problem, an invariant, called the resurgence, was introduced by Bocci and Harbourne, ?(I) = sup{ r/s : I^(r) is not contained in I^s}. Computing this invariant or obtaining optimal bounds, will give an effective answer to the containment problem stated above. It was shown by Hochster and Huneke that if I is a radical ideal of big heigth h in a regular ring, then I^(hn) ?I^n for all n. Huneke conjectured that if P is a height 2 prime ideal in a 3 dimensional regular ring, then P^(3) ?P^2. Harbourne extended this conjecture that if I is a radical ideal of big height h in a regular ring, then I^(hn-h+1) ? I^n for all n. There are counterexamples to this conjecture, but has been proved for large classes of ideals, for example, squarefree monomial ideals in polynomial rings over a field. Grifo modified this conjecture and asked if the containment I^(hn-h+1) ? I^n is true for all large values of n. This is known as stable Harbourne conjecture. As of today, there are no counterexamples to this conjecture. I propose to investigate the stable Harbourne conjecture as well as study the resurgence for several important classes of ideals. I also propose a new invariant: If R is a Noetherian ring of characteristic p and I an ideal in R, then ?*(I) := sup{ r/s : I^(r) is not contained in (I^s)*}, where J* denote the tight closure of an ideal J. I also propose to investigate the relation between ?(I) and ?*(I). |