On the containment problem

The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area of ongoing active research. We conclude the note with a list of potential promising paths of further research.Comment: 13 pages, 1 figur

On Parameterizations of plane rational curves and their syzygies

Let $C$ be a plane rational curve of degree $d$ and $p:\tilde C \rightarrow C$ its normalization. We are interested in the splitting type $(a,b)$ of $C$, where $\mathcal{O}_{\mathbb{P}^1}(-a-d)\oplus \mathcal{O}_{\mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)\subset K[s,t]$, and $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2k\leq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $\mathbb{P}^{k+1}$.Comment: 12 Page

Osculating spaces to secant varieties

We generalize the classical Terracini's Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of the $d$-Veronese embedding of the projective 3-space

Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals

We study symbolic powers of bi-homogeneous ideals of points in the Cartesian product of two projective lines and extend to this setting results on the effect of points fattening obtained by Bocci, Chiantini and Dumnicki, Szemberg, Tutaj-Gasi\'nska. We prove a Chudnovsky-type theorem for bi-homogeneous ideals and apply it to classification of configurations of points with minimal or no fattening effect. We hope that the ideas developed in this project will find further algebraic and geometric applications e.g. to study similar problems on arbitrary surfaces.Comment: 12 pages, notes from a workshop on linear series held in Lanckoron