Let (R,m_R) be a d-dimensional, excellent, normal local ring. A divisorial filtration {I_n} is determined by a divisor D on a normal scheme X determined by blowing up an ideal on R, so that I_n are the global sections of nD. Associated to an m_R-primary divisorial filtration, we have the Hilbert function f(n)=\lambda_R(R/I_n), where \lambda_R is the length of an R-module. We discuss how close or far this function is from being a polynomial, focusing on examples which are constructed and analyzed geometrically.

In this talk, we consider a finite rational map determined by a linear system with base locus. The degree of such map has been studied in many situation, for instance, the degree of Gauss map of theta divisors. In principal, this degree can be computed by Segre class of the base locus. In practice, one can use Vogel's cycle to give an estimation. Associated to the base locus, one can define distinguished subvarieties, which has been used to the study of geometric Nullstellensatz by Ein-Lazarsfeld. We discuss how distinguished subvarieties and their coefficients can be used to estimate the degree of finite rational map. This is a joint work with Yilong Zhang.

Given two relatively prime positive integers, p < q, Kunz and Waldi defined a class of numerical semigroups which we denote by KW(p, q) consisting of semigroups of embedding dimension n and type n−1 and multiplicity p by filling in the gaps of the semigroup < a, b >. We study these semigroups, give a criterion for these in terms of principal matrices or their critical binomials and generalize the notion to KW(p, q, w) and prove some results and questions. We will discuss their resolutions and Betti Numbers. Most of this is a joint work with Srishti Singh.