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Symbolic Powers in Algebra and Geometry
Given a set of points in space, the symbolic powers of their vanishing ideal describe the polynomial functions that vanish at those points and have partial derivatives that, up to a certain order, also vanish at those points. The symbolic powers and ordinary powers of an ideal are closely related. This thesis considers two approaches to comparing symbolic and ordinary powers of ideals: containment questions and the symbolic defect. Given an ideal I, the containment problem asks for which integers m and r is the symbolic m-th power of I contained in the ordinary r-th power of I. It is known that the symbolic hr-th power of I is contined in the ordinary r-th power, where h is the maximum among the heights of associated primes of I. We call an ideal containment-tight when there is some r such that this containment is the best possible (i.e. (hr-1)-st symbolic power of I is not contained in the r-th ordinary power ). Most previously known examples of containment-tight ideals arise as the defining ideals of singular loci of reflection arrangements. This thesis classifies which reflection arrangements have containment-tight ideals defining their singular loci where the symbolic third power is not contained in the ordinary second power. Given an ideal I, the m-th symbolic defect of I (denoted sdefect(I,m)) counts how many generators must be added to the m-th power of I to get the symbolic m-th power. This thesis studies the behavior of sdefect(I,m) for large values of m, and concludes that when the symbolic Rees algebra of I is Noetherian sdefect(I,m) is eventually quasi-polynomial.
Drabkin, Benjamin, "Symbolic Powers in Algebra and Geometry" (2020). ETD collection for University of Nebraska - Lincoln. AAI27956145.