CATALISANO , Maria Virgina

We consider the $k$-osculating varieties $O_{k,d}$ to the Veronese $d-$uple embeddings of $\PP 2$. By studying the Hilbert function of certain zero-\-dimensional schemes $Y\subset \PP 2$, we find the dimension of $O^s_{k,d}$, the $(s-1)^{th}$secant varieties of $O_{k,d}$, for $3 \leq s\leq 6$ and $s=9$, and we determine whether those secant...

In this paper we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of $\mathbb{P}^n\times \mathbb{P}^1$ via the section of the sheaf $\mathcal{O}(a,b)$ for any $n,a,b\in \mathbb{Z}^+$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann...

Let $X^{(n,m)}_{(1,d)}$ denote the Segre\/-Veronese embedding of $\PP n \times \PP m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$...

A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\PP n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of...

We consider the varieties $O_{k,n.d}$ of the k-osculating spaces to the Veronese varieties, the $d-$uple embeddings of $\PP n$; we study the dimension of their higher secant varieties. Via inverse systems (apolarity) and the study of certain spaces of forms we are able, in several cases, to determine whether those secant varieties are defective or not.

For any irreducible non-degenerate variety $X \subset \mathbb{P}^r$ , we relate the dimension of the $s$-th secant varieties of the Segre embedding of $\mathbb{P}^k\times X$ to the dimension of the $(k,s)$-Grassmann secant variety $GS_X(k,s)$ of $X$. We also give a criterion for the $s$-identifiability of $X$.