Search

Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$ in $n$ variables, and given an integer $\ell>1$ and a partition $\lambda$ of $d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions of the $\ell$-secant varieties $\sigma_\ell ({\mathbb X}_{n-1,\lambda})$ for the subvariety ${\mathbb X}_{n-1,\lambda} \subset V$ of hypersurfaces whose defining forms have a factorization into forms of degrees $d_1,\ldots,d_r$. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of $\sigma_\ell({\mathbb X}_{n-1,\lambda})$ for any choice of parameters $n,\ell$ and $\lambda$. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., $r=2$), we also relate this problem to a conjecture by Fr\"oberg on the Hilbert function of an ideal generated by general forms., Comment: 48 pages; corrected a typo in the statement of Proposition 7.2 and added short explanation. Appeared in J. Algebra

International audience; 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 ${m+d \choose d}$ is a multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.

Let X be a closed subscheme and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d)=min{hp(P^n,d),hp(X,d)} for every non-negative integer d. We show that if X consists of a plane and generic lines, then X has bipolynomial Hilbert function. We also conjecture that generic configurations of non-intersecting linear spaces have bipolynomial Hilbert function.

A subspace arrangement in a vector space is a finite collection of vector subspaces. Similarly, a configuration of linear spaces in a projective space is a finite collection of linear subspaces. In this paper we study the degree 2 part of the ideal of such objects. More precisely, for a generic configuration of linear spaces L we determine HF(L,2), i.e. the Hilbert function of L in degree 2., Comment: 24 pages, revised version with notation problem solved in the last section

Let V_n be the Segre embedding of (P^1) x ... X (P^1) (n times). We prove that the higher secant varieties, \sigma_s(V_n), always have the expected dimension, except for \sigma_3(V_4), which is of dimension 1 less than expected.

A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\mathbb{P}^n$) 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 $\mathbb{P}^n$., Comment: 23 pages

In this paper we compute the dimension of all the $s^{th}$ higher secant varieties of the Segre-Veronese embeddings $Y_{\underline d}$ of the product $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$ in the projective space $\mathbb{P}^N$ via divisors of multi-degree $\underline d = (a, b, c) (N = (a + 1)(b+1)(c+1)-1)$. We find that $Y_{\underline d}$ has no deficient higher secant varieties, unless $\underline d$ = (2, 2, 2) and $s$ = 7, or $\underline d$ = (2h, 1, 1) and $s$ = 2h + 1, with defect 1 in both cases.

We consider the $k$-osculating varieties $O_{k,d}$ to the Veronese $d$-uple embeddings of $\mathbb{P}^2$. By studying the Hilbert function of certain zero-dimensional schemes $Y \subset \mathbb{P}^2$, 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 varieties are defective or not.

We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} = {\Bbb P}^{n_1}\times ...\times {\Bbb P}^{n_t}$ embedded the morphism given by ${\cal O}_{\Bbb X}({a_1,...,a_t})$. We call it a {\it Segre-Veronese variety} and the embedding a {\it Segre-Veronese embedding}. In several cases (e.g. for 3 factors {\Bbb P}^{1}, or when $s$ is small) we show that $X^s$ has the expected dimension except for a few cases. A list of examples of defective $X^s$'s is given., Comment: 12 pages, plain TEX

In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space to translate the problem into that of finding the dimension of a graded piece of a " fat" ideal in the exterior algebra. Among other things, our methods prove that (apart from the chordal varieties to the Grassmannians of lines) all the chordal varieties of Grassmannians have the expected dimension. We also discuss some Grassmannians with deficient secant varieties, giving new proofs that G(3,7) and G(3,9) are of this type. We also find a new Grassmannian, namely G(4,8) with deficient secant varieties., Comment: 16 pages, plain tex, corrected some typos and misunderstanding of a conjecture

In this paper we find an algorithm which computes the Hilbert function of schemes $Z$ of "fat points" in $\PP3$ whose support lies on a rational normal cubic curve $C$. The algorithm shows that the maximality of the Hilbert function in degree $t$ is related to the existence of fixed curves (either $C$ itself or one of its secant lines) for the linear system of surfaces of degree $t$ containing $Z$., Comment: 20 pages, plain TEX, no figures