Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to appea

Anthropogenic impact on a pregnant Cuvier’s beaked whale (Ziphius cavirostris) stranded in Brazil

Background: Because of their usually cryptic behaviour, most knowledge on the biology of beaked whales are from records of stranded animals. Although the Cuvier’s beaked whale (Ziphius cavirostris) is the best known species of the ziphiidae family, little information on its reproduction is available. Results: Here we report on the stranding of a dead pregnant female with clear signs of anthropogenic impact, including the presence of a fishing artefact in the stomach. Conclusions: The region of the stranding (north-eastern coast of Brazil) is an area of increasing interest for oil and gas exploitation. Conservation concerns may arise from findings such as the one presented and discussed here