Symbolic powers in weighted oriented graphs
Abstract
Let D be a weighted oriented graph with the underlying graph G when vertices with non-trivial weights are sinks and I(D),I(G) be the edge ideals corresponding to D and G, respectively. We give an explicit description of the symbolic powers of I(D) using the concept of strong vertex covers. We show that the ordinary and symbolic powers of I(D) and I(G) behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of I(D) for certain classes of weighted oriented graphs. When D is a weighted oriented odd cycle, we compute reg(I(D)(s)/I(D)s) and prove regI(D)(s)≤regI(D)s and show that equality holds when there is only one vertex with non-trivial weight.
Communicated by J. McCullough