Symbolic powers of monomial ideals and ideals of points
The symbolic power I(n) of an ideal I, which is related to the primary decomposition of the ordinary power I^n, has been of interest to many mathematicians, especially in the field of commutative algebra and algebraic geometry for various reasons, one of those is the fact that symbolic powers encode vanishing condition of functions on geometric objects. Monomial ideals and ideals of points are among the most important classes of ideals in polynomial rings with many applications in other fields of study. While questions regarding polynomials in the symbolic powers of ideals of points are equivalent to questions in polynomial interpolation problem, various questions regarding symbolic powers of monomial ideals and their invariant have interesting connection to questions in graph theory, convex (polyhedral) geometry, and integer programming. In this dissertation, we focus on the problem of comparing the symbolic powers and ordinary powers of an ideal and studying the closely related algebraic invariant. In particular, we prove stable Harbourne-Huneke containment between symbolic powers and ordinary powers of ideals of general points and derive Chudnovsky's and Demailly's Conjecture for those ideals. On the other hand, we use combinatorial data of the Newton-Okounkov body to study the Noetherian property and related algebraic invariant of the Rees algebra of a graded family, which is a generalization of family of symbolic powers, of monomial ideals.