36 research outputs found
On the Hilbert function of general fat points in
We study the bi-graded Hilbert function of ideals of general fat points with
same multiplicity in . 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 embedded in bi-degree
Let denote the Segre-Veronese embedding of via the sections of the sheaf . We study
the dimensions of higher secant varieties of and we prove
that there is no defective secant variety, except possibly for
values of . Moreover when is multiple of , the
secant variety of has the expected dimension for
every .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 .
A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of (the -uple embedding of \PP n) have the expected dimension, with few known exceptions. We study here the same problem for , the tangential variety to , and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for . Moreover. we prove that it holds for any if it holds for . Then we generalize to the case of , the -osculating variety to , proving, for , a conjecture that relates the defectivity of to the Hilbert function of certain sets of fat points in \PP n
Waring-like decompositions of polynomials - 1
Let be a homogeneous form of degree in variables. A Waring
decomposition of is a way to express as a sum of 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 -space which
are -symmetric, i.e. they intersect any curves thru the point with length
. We show that the maximal length for such a scheme is (-superfat
points) and we study properties of such schemes, in particular for . We
also study varieties defined by such schemes on Veronese and Segre-veronese
varieties.Comment: 25 pages, 11 figure