Let I be an m-primary ideal of a Noetherian local ring (R, m) of positive dimension. The coefficient e 1 (A) of the Hilbert polynomial of an I-admissible filtration A is called the Chern number of A.The Positivity Conjecture of Vasconcelos for the Chern number of the integral closure filtration {I n } is proved for a 2-dimensional complete local domain and more generally for any analytically unramified local ring R whose integral closure in its total ring of fractions is Cohen-Macaulay as an R-module. It is proved that if I is a parameter ideal then the Chern number of the I-adic filtration is non-negative. Several other results on the Chern number of the integral closure filtration are established, especially in the case when R is not necessarily Cohen-Macaulay.

Abstract. Let R be an analytically unramified local ring with maximal ideal m and d = dim R > 0. If R is unmixed, then e 1 I (R) ≥ 0 for every m-primary ideal I in R, where e 1 I (R) denotes the first coefficient of the normal Hilbert polynomial of R with respect to I. Thus the positivity conjecture on e 1 I (R) posed by Wolmer V. Vasconcelos is settled affirmatively.

Let G be a graph and I = I(G) be its edge ideal. When G is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of I and compute the Waldschmidt constant. When G is complete graph then we describe the generators of the symbolic powers of I and compute the Waldschmidt constant and the resurgence of I. Moreover for complete graph we prove that the Castelnuovo-Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.

No abstract

Let [Formula: see text] be a weighted oriented graph with the underlying graph [Formula: see text] when vertices with non-trivial weights are sinks and [Formula: see text] be the edge ideals corresponding to [Formula: see text] and [Formula: see text] respectively. We give an explicit description of the symbolic powers of [Formula: see text] using the concept of strong vertex covers. We show that the ordinary and symbolic powers of [Formula: see text] and [Formula: see text] behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of [Formula: see text] for certain classes of weighted oriented graphs. When [Formula: see text] is a weighted oriented odd cycle, we compute [Formula: see text] and prove [Formula: see text] and show that equality holds when there is only one vertex with non-trivial weight.