On the Hilbert function of general fat points in P1×P1\mathbb{P}^1 \times \mathbb{P}^1

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.Comment: 25 pages; minor changes (Remark 1.7 added and Example 3.13 improved

Higher secant varieties of Pn×Pm\mathbb{P}^n \times \mathbb{P}^m embedded in bi-degree (1,d)(1,d)

Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ 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 ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.Comment: 8 page

Progress on the symmetric Strassen conjecture

Let F and G be homogeneous polynomials in disjoint sets of variables. We prove that the Waring rank is additive, thus proving the symmetric Strassen conjecture, when either F or G is a power, or F and G have two variables, or either F or G has small rank

Secant varieties to osculating varieties of Veronese embeddings of PnP^n.

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 Alexander-Hirschowitz theorem, for $n\leq 9$. Moreover. we prove that it holds for any $n,d$ if it holds for $d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity of $\sigma_s(O_{k,n,d})$ to the Hilbert function of certain sets of fat points in \PP n

Waring-like decompositions of polynomials - 1

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.Comment: 12 pages; Section 5 added in this versio

Superfat points and associated tensors

We study the 0-dimensional schemes supported at one point in $n$-space which are $m$-symmetric, i.e. they intersect any curves thru the point with length $m$. We show that the maximal length for such a scheme is $m^n$ ($m$-superfat points) and we study properties of such schemes, in particular for $n=2$. We also study varieties defined by such schemes on Veronese and Segre-veronese varieties.Comment: 25 pages, 11 figure