Integrality, Blowup Algebras and Multiplicities

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Passing from an affine algebra or an ideal to the integral closureis a fundamental operation. The investigator intends to examinecomputational aspects of this process. While there exist algorithmsfor computing the integral closure of an affine domain, little isknown about the complexity of such calculations. To address thisproblem the investigator intends to find estimates for the numberof algebra generators of the integral closure and for the degreesof these generators. In the case of ideals on the other hand, onestill has no efficient method for computing integral closures. Inview of this lack of an algorithm, the investigator intends toexpress, or at least approximate, the integral closure of an idealby a sum of colon ideals that are more readily accessible. The coreof an ideal is a counterpart of the integral closure: it encodesinformation about all possible reductions of a given ideal, i.e.,all ideals over which the given ideal is integral. The investigatorintends to work on a conjectural formula that, if proved, would givean explicit expression for the core of a large class of ideals.Continuing the somewhat computational theme the investigator proposesto establish bounds for the Castelnuovo-Mumford regularity ofembedded projective varieties in terms of their defining equations.The known results require strong assumptions on the singularities ofthe variety which the investigator hopes to weaken. In this vein heintends to show that rational singularities persist under genericlinkage of ideals in the linkage class of a complete intersection.In continuation of his earlier work on blowup algebras theinvestigator plans to study the rings representing special fibers ofblowups. These `special fiber rings' describe, for instance, imagesof rational maps, including secant varieties and Gauss images. Theinvestigator intends to study the Cohen-Macaulayness of special fiberrings, estimate their depth and compute multiplicities. Suitabledepth estimates would have a bearing on a conjecture by Mazur thatoriginates in Wiles' work on deformations of Galois representations.Formulas for the multiplicity on the other hand would lead to anextension of the Teissier-Pluecker formula for the degrees of certaindual varieties.The investigator works in the area of Mathematics called "Commutative Algebra," which deals with the qualitative study of systems of polynomial equations in several variables. Such systems arise in numerous applications outside mathematics. Over the past two decades commutative algebraists have become more concerned with computational aspects, thereby emphasizing connections to applied areas such as computer algebra, robotics, cryptography and coding theory. This project addresses both theoretical and computational aspects of commutative algebra.