Cores, regularity and principal ideal theorems

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The core of an ideal in a commutative ring encodes information about allpossible reductions of the ideal. It also has a close connection toBriancon-Skoda type theorems and to a conjecture by Kawamata aboutsections of line bundles. The proposer intends to further explore thisinterplay by studying the relation between cores and adjoints ormultiplier ideals. Having worked on a formula for the core inequicharacteristic zero, he wishes to obtain a similar explicit expressionin positive characteristic, where the shape of the core is markedlydifferent. Likewise he would like to find a combinatorial description forthe core of monomial ideals. The investigator plans to continue his workon blowup algebras of ideals, most notably of zero-dimensional ideals inregular local rings. He asks whether the quasi-Gorenstein property of theextended Rees algebra implies the Gorensteinness of the associated gradedring. He also suggests that the normality or Cohen-Macaulayness of thespecial fiber ring of a Gorenstein ideal may force the ideal to be acomplete intersection. On a more computational note, he addresses theproblem of constructing the integral closure of algebras, in particular ofRees algebras. Passing to the integral closure of the Rees algebra of anideal is the first step towards resolution of singularities and the onlyknown general method for computing the integral closure of the ideal. Theproposer wishes to estimate the complexity of this process by finding boundson the number of generators of the integral closure, the degrees of thegenerators and the number of steps required in the computation. As anothermeasure of complexity he plans to study the Castelnuovo-Mumford regularityof powers and symmetric powers of homogeneous ideals having dimension atmost one. He expects that estimates on the regularity do not only persistwhen the ideal is raised to powers, but that they actually improve. Similarimproved bounds for the regularity of symmetric powers would help findingthe equations of Rees algebras and thereby lead to efficient algorithms inelimination theory. The proposer also intends to continue his work ongeneralized principal ideal theorems. The goal is to bound the codimensionand prove connectedness properties for degeneracy loci of maps of modulesthat are not necessarily free; here one has to assume that the maps are not`too generic'. The investigator proposes a weak version of this conditionby introducing a notion of ampleness for modules over local rings. Hehopes to prove principal ideal theorems that only require the weakerassumption, thus generalizing the known results in both local algebra andprojective geometry.The investigator works in Commutative Algebra, an area concerned with thequalitative study of systems of polynomial equations in several variables.Such systems arise in numerous applications outside of mathematics. Overthe past two decades commutative algebraists have become increasinglyinterested in computational aspects, thereby emphasizing connections toapplied areas such as computer algebra, robotics, cryptography and codingtheory. This investigator's research too has a strong computationalcomponent.Part of the project involving the collaboration with mathematicians in Brazil is funded by the NSFOffice of International Science and Engineering