Limit behavior of the rational powers of monomial ideals

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

The Theatre of the Senses: Introductory Words and a Short Conversation for the Symposium

What happens to us when we enter a Teatro de los Sentidos (Theatre of the Senses) experience? Why do we need to play? Where does our need to play come from? Why is sensory theatre relevant today? For what strange reasons do humans like to play getting lost and finding themselves in the dark? I want to share with you the reasons why Teatro de los Sentidos has been significant to me ever since I became aware of inventing myself and inventing it in my childhood games, imagining forbidden labyrinths in Colombian coffee plantations, until today, at my eighty years of age. It is clear that a lot of primal and parallel knowledge has developed over time in all cultures. Myths, celebrations, imagination, poetry, symbolic powers ... resonate differently, each according to historical circumstances. Let us ask ourselves together: what potential does sensorial theatre have today? (Enrique Vargas: Some questions before the symposium.)

Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote the corresponding edge ideal in a polynomial ring over a field [Formula: see text]. In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

Symbolic powers of cover ideals of graphs and Koszul property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

Symbolic powers in weighted oriented graphs

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.